INTERNATIONAL GONGRESS

82 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"> \begin{matrix} (2.3) & {| P_{H, Î¾} (Ï†) |}^{2} = {| S_{Ï€} |}^{âˆ’ 1} Î” \frac{L (\frac{1}{2}, {Ï€}_{E})}{L (1, Ï€, A d)} \underset{v}{âˆ} P_{H, Î¾, v}^{â™®} ({Ï†}_{v}, {Ï†}_{v}) \end{matrix}

where

Δ = \prod_{i = 1}^{n} L (i, η_{E / F}^{i})

and

L (s, π, A d) = \prod_{v} L (s, π_{v}, A d)

denotes the completed adjoint

L

-function of

π

Note that at a formal level, that is, formally expanding

L

-functions as Euler products outside the range of convergence, the above formula can be rewritten in the more compact way as

\begin{matrix} (2.4) & {| P_{H, ξ} (φ) |}^{2} = {| S_{π} |}^{- 1} \prod_{v}^{'} P_{H, ξ, v} (φ_{v}, φ_{v}) \end{matrix}

where the prime symbol on the product sign indicates that it is not convergent and has to be suitably reinterpreted "in the sense of

L

-functions" as identity (2.3).

Thanks to the work of many authors that we are going to summarize in the next sections, it is now relatively easy to state the current status on these two conjectures:

Theorem 2.1. Both Conjectures 2.1 and 2.2 hold in full generality.

The rest of this paper is devoted to reviewing the long series of works leading to the above theorem. They all stem from a strategy originally proposed by Jacquet and Rallis [32] of comparing two relative trace formulae. Let us mention here that there has actually been other fruitful approaches to the global Gan-Gross-Prasad conjecture among which one of the most notable has been the method pioneered by Ginzburg-Jiang-Rallis [25] using automorphic descent and that has recently seen much development with the work [33] of Jiang and L. Zhang proving in full generality the implication (2)

\Rightarrow

(1) of Conjecture 2.1.

2.2. The approach of Jacquet-Rallis

In [32], Jacquet and Rallis have proposed a way to attack the Gan-Gross-Prasad conjecture for unitary groups through a comparison of relative trace formulae. They only consider the case where

\dim (W) = \dim (V) - 1

(in which case

H = U (W)

and the character

ξ

is trivial) and we assume throughout that this condition is satisfied. The global relative trace formulae considered here are powerful analytic tools introduced originally by Jacquet and that relate automorphic periods to more geometric distributions known as relative orbital integrals.

Let us be more specific in the case at hand. For

f \in C_{c}^{\infty} (G (A_{F}))

, a global test function, we let

K_{f} (x, y) = \sum_{γ \in G (F)} f (x^{- 1} γ y), x, y \in G (F) ∖ G (A_{F})

be its automorphic kernel which describes the operator

R (f)

of right convolution by

f

on the space of automorphic forms. The first trace formula introduced by Jacquet and Rallis is formally obtained by expanding the (usually divergent) expression

\begin{matrix} (2.5) & J (f) = \int_{[H] \times [H]} K_{f} (h_{1}, h_{2}) d h_{1} d h_{2} \end{matrix}

in two different ways. More precisely, but still at a formal level, this distribution can be expanded as

\begin{matrix} (2.6) & \dots + \sum_{δ \in H (F) ∖ G_{r s} (F) / H (F)} O (δ, f) = J (f) = \sum_{φ \in A_{cusp} (G)} P_{H} (R (f) φ) \bar{P_{H} (φ)} + \dots \end{matrix}

where the right sum runs over an orthonormal basis for the space of cuspidal automorphic forms whereas the left sum is indexed by the so-called regular semisimple double cosets of

H (F)

G (F)

. Here, an element

δ \in G

is called (relatively) regular semisimple if its stabilizer under the

H \times H

-action is trivial and the corresponding orbit is (Zariski) closed. We denote by

G_{r s}

the nonempty Zariski open subset of regular semisimple elements and for

δ \in G_{r s} (F)

O (δ, f) = \int_{H (A_{F}) \times H (A_{F})} f (h_{1} δ h_{2}) d h_{1} d h_{2}

denotes the corresponding relative orbital integral of

f

δ

. The left suspension points in (2.6) represent the remaining contributions from singular orbits whereas the right suspension points indicate the contribution of the continuous spectrum (both of which are somehow responsible for the divergence of the original expression (2.5)).

The second trace formula introduced by Jacquet and Rallis has to do with the following triple of groups:

\begin{aligned} H_{1} & = {Res}_{E / F} {G L}_{n, E} ↪ G^{'} = {Res}_{E / F} {G L}_{n + 1, E} \times {Res}_{E / F} {G L}_{n, E} ↩ H_{2} \\ = {G L}_{n + 1, F} \times {G L}_{n, F}, \end{aligned}

where

n = \dim (W)

, the first embedding is the diagonal one and the second embedding is the natural one. Note that

G^{'}

is the group on which the base-change

π_{E}

"lives." For

f^{'} \in C_{c}^{\infty} (G^{'} (A_{F}))

, we write (again formally)

\begin{matrix} (2.7) & I (f^{'}) = \int_{[H_{1}] \times [H_{2}]} K_{f^{'}} (h_{1}, h_{2}) η (h_{2}) d h_{1} d h_{2} \end{matrix}

where

K_{f^{'}}

is the automorphic kernel of

f^{'}

and

η : [H_{2}] \to {\pm 1}

is the automorphic character defined by

η (g_{n}, g_{n + 1}) = η_{E / F} {(\det g_{n})}^{n + 1} η_{E / F} {(\det g_{n + 1})}^{n}

. This formal distribution can be analogously expanded as

\begin{matrix} (2.8) & \dots + \sum_{γ \in H_{1} (F) ∖ G_{r s}^{'} (F) / H_{2} (F)} O_{η} (γ, f^{'}) = I (f^{'}) = \sum_{φ \in A_{cusp} (G^{'})} P_{H_{1}} (R (f^{'}) φ) \bar{P_{H_{2}, η} (φ)} + \dots \end{matrix}

where

G_{r s}^{'}

stands for the open subset of regular and semisimple elements under the

H_{1} \times H_{2}

action, the relative orbital integrals are given by

O_{η} (γ, f^{'}) = \int_{H_{1} (A_{F}) \times H_{2} (A_{F})} f^{'} (h_{1} γ h_{2}) η (h_{2}) d h_{1} d h_{2}

and

P_{H_{1}}, P_{H_{2}, η}

denote the automorphic period integrals over

[H_{1}]

and

[H_{2}]

twisted by

η

, respectively.

The discussion so far is, of course, oversimplifying and ignoring serious analytical and convergence issues (we will come back to this later). However, as a motivation for considering this relative trace formula on

G^{'}

, we have the following results on automorphic periods:

The period $P_{H_{1}}$ is a Rankin-Selberg period studied by Jacquet-PiatetskiiShapiro-Shalika that essentially represents the central value $L (\frac{1}{2}, Π)$ on $Π ↪ A_{cusp} (G^{'})$ ;
The period $P_{H_{2}, η}$ was studied by Rallis and Flicker who have shown that it detects exactly the cuspidal automorphic $Π$ 's that come by base-change from $G$ (i.e., it is nonzero precisely on those cuspidal representations of the form $π_{E}$ , for $π$ a cuspidal automorphic representation of $G$ ).

Thus, on a very formal and sketchy sense, the Gan-Gross-Prasad conjecture implies that the spectral sides of

I (f^{'})

should somehow "match" that of

J (f)

. The idea of Jacquet and Rallis was to make precise the existence of such a comparison, from which the global GanGross-Prasad conjecture was eventually to be deduced, by equalling the geometric sides term by term. As a first step, they define a correspondence of orbits, which here takes the form of a natural embedding between regular semisimple cosets

\begin{matrix} (2.9) & H (k) ∖ G_{r s} (k) / H (k) ↪ H_{1} (k) ∖ G_{r s}^{'} (k) / H_{2} (k), δ \mapsto γ \end{matrix}

for every field extension

k / F

. Using this correspondence, they then introduced a related notion of local transfer (or matching): for a place

v

F

, two test functions

f_{v} \in C_{c}^{\infty} (G_{v})

and

f_{v}^{'} \in C_{c}^{\infty} (G_{v}^{'})

are said to be transfers of each other (simply denoted by

f_{v} \leftrightarrow f_{v}^{'}

for short) if for every

δ \in H (F_{v}) ∖ G_{r s} (F_{v}) / H (F_{v})

we have an identity

\begin{matrix} (2.10) & O (δ, f_{v}) = Ω_{v} (γ) O_{η_{v}} (γ, f_{v}^{'}) \end{matrix}

where

γ \in H_{1} (F_{v}) ∖ G_{r s}^{'} (F_{v}) / H_{2} (F_{v})

is the image of

δ

by the above correspondence,

O (δ, f_{v})

and

O_{η_{v}} (γ, f_{v}^{'})

are local relative orbital integrals defined in the same way as their global counterparts (replacing in the domain of integration, adelic groups by the corresponding local groups) and

γ \mapsto Ω_{v} (γ)

is a certain transfer factor which in particular has the effect of making the right-hand side above

H_{1} (F_{v}) \times H_{2} (F_{v})

-invariant in

γ

As in the usual paradigm of endoscopy, to make this notion useful and allow for a global comparison we basically need two local ingredients: first the existence of local transfer (i.e., for every

f_{v} \in C_{c}^{\infty} (G_{v})

there exists

f_{v}^{'} \in C_{c}^{\infty} (G_{v}^{'})

such that

f_{v} \leftrightarrow f_{v}^{'}

and conversely, every

f_{v}^{'}

admits a transfer

f_{v}

) and then a fundamental lemma (saying, at least, that

1_{G (O_{v})} \leftrightarrow 1_{G^{'} (O_{v})}

for almost all

v)

3. COMPARISON: LOCAL TRANSFER AND FUNDAMENTAL LEMMA

A first breakthrough on the Jacquet-Rallis approach to the Gan-Gross-Prasad conjecture was made in [57] by Wei Zhang who proved the existence of the local transfer at all non-Archimedean places. His strategy for doing so roughly goes as follows:

The first step is to reduce to a statement on Lie algebras using some avatar of the exponential map (also known as Cayley map): we are then left with proving the existence of a similar transfer between the orbital integrals for the adjoint action of $U (W_{v})$ on $u (V_{v}) = Lie (U (V_{v}))$ and for the adjoint action of ${G L}_{n} (F_{v})$ on $g_{n + 1} (F_{v})$ .
Then, a crucial ingredient in Zhang's proof is to show that the transfer at the Lie algebra level essentially commutes (i.e., up to some explicit multiplicative
constants) with 3 different partial Fourier transforms $F_{1}, F_{2}$ , and $F_{3}$ that can naturally be defined on the two spaces $C_{c}^{\infty} (u (V_{v})), C_{c}^{\infty} ({g l}_{n + 1} (F_{v}))$ . One of them, that we will denote by $F_{1}$ , is the Fourier transform with respect to "the last row and column" on ${g l}_{n + 1} (F_{v})$ or $u (V_{v})$ when realizing the latter in matrix form using a basis adapted to the decomposition $V_{v} = W_{v} \oplus W_{v}^{⊥}$ . (Recall that we are assuming that $\dim (W_{v}^{⊥}) = 1$ .) For this, Zhang develops some relative trace formulae for the aforementioned actions on ${g l}_{n + 1} (F_{v})$ and $u (V_{v})$ and combines them with a clever induction argument.
Finally, the proof of the existence of transfer on Lie algebras is obtained by combining the second step with a certain uncertainty principle due to Aizenbud [1], which allows reducing the construction of the transfer to functions that are supported away from the relative nilpotent cones (i.e., the set of elements whose orbit closure contains an element of the center of the Lie algebra), as well as a standard descent argument whose essence goes back to Harish-Chandra.

It is noteworthy to mention that this result was subsequently extended, following the same strategy, by H. Xue [53] to Archimedean places, although the final result there is slightly weaker. (More precisely, Xue was only able to show that a dense subspace of test functions admit a transfer but also observed that it is sufficient for all expected applications.)

The Jacquet-Rallis fundamental lemma for its part, was proven earlier by Yun [55] in the case of fields of positive characteristic following and adapting the geometriccohomological approach based on Hitchin fibrations that was developed by

N g ô

in the context of the endoscopic fundamental lemma. This result was then transferred to fields of characteristic zero, but of sufficiently large residual characteristic, using model-theoretic techniques by Julia Gordon in the appendix of [55].

Later, in [14], I found a completely new and elementary proof of this fundamental lemma. The argument, despite that of Gordon-Yun, works directly in characteristic zero and is purely based on techniques from harmonic analysis. Thus, we have:

Theorem 3.1 (Yun-Gordon, Beuzart-Plessis). Let

v

be a place of

F

of residue characteristic not 2 that is unramified in

E

and assume that the Hermitian spaces

W_{v}, W_{v}^{⊥}

both admit self-dual lattices

L_{v}^{W}

and

L_{v}^{W^{⊥}}

. Set

L_{v} = L_{v}^{W} \oplus L_{v}^{W^{⊥}}

(a self-dual lattice in

V_{v}

) and

K_{v} = {Stab}_{G_{v}} (L_{v} \times L_{v}^{W})

for the stabilizer in

G_{v} = U (V_{v}) \times U (W_{v})

of the lattices

L_{v}

and

L_{v}^{W}

(a hyperspecial compact subgroup of

G_{v}

). Then, setting

K_{v}^{'} = {GL}_{n + 1} (O_{E_{v}}) \times

{GL}_{n} (O_{E_{v}})

, we have

1_{K_{v}} \leftrightarrow 1_{K_{v}^{'}}

More precisely, in [14] I proved a Lie algebra analog of the Jacquet-Rallis fundamental lemma (of which the original statement can easily be reduced; at least in residual characteristic not 2) stating that the relative orbital integrals of

1_{u (L_{v})}

match those of

1_{g l_{n + 1}} (O_{F_{v}})

in a suitable sense (where

u (L_{v})

denotes the lattice in

u (V_{v})

stabilizing

L_{v}

). The argument is based on a hidden SL(2) symmetry involving a Weil representation. More specifically, we consider the Weil representations of

SL (2, F_{v})

associated to the quadratic form

q

sending a
matrix of size

n + 1

X = (\begin{array}{ll} A & b \\ c & λ \end{array})

either in

{g l}_{n + 1} (F_{v})

or in

u (V_{v})

, to

q (X) = c b

(where here,

A

is a square-matrix,

b

is a column vector, and

c

a row vector all of size

n

). Using the aforementioned result of Zhang that the transfer commutes with the partial Fourier transform

F_{1}

, it can be shown that these representations descend to spaces of relative orbital integrals on

C_{c}^{\infty} (u (V_{v}))

and

C_{c}^{\infty} ({g l}_{n + 1} (F_{v}))

and coincide on their intersections (identifying the spaces of regular semisimple orbits through a correspondence similar to (2.9)). Consider then the difference

Φ : X \in u {(V_{v})}_{r s} / U (W_{v}) \mapsto O (X, 1_{u (L_{v}^{V})}) - ω_{v} (Y) O_{η_{v}} (Y, 1_{g I_{n + 1} (O_{F_{v}})})

where

u {(V_{v})}_{rs}

denotes the Lie algebra analog of the relative regular semisimple locus,

Y

is the image of

X

by a correspondence of orbits

u {(V_{v})}_{r s} / U (W_{v}) ↪ {g l}_{n + 1} {(F_{v})}_{r s} / {G L}_{n} (F_{v})

similar to (2.9) and

ω_{v} (Y)

is the Lie algebra counterpart of the transfer factor. The fundamental lemma then states that

Φ

is identically zero. The proof proceeds roughly in three steps:

First, we show that $Φ (X) = 0$ for $| q (X) | ⩾ 1$ . When $| q (X) | = 1$ , this requires an inductive argument on $n$ . Moreover, this vanishing can be reformulated by saying that $Φ$ is fixed by the subgroup $(\begin{array}{cc} 1 & p_{F_{v}}^{- 1} \\ 0 & 1 \end{array})$ through the Weil representation.
Secondly, we remark that $Φ$ is also fixed by $w = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ . This comes from the fact that the action of $w$ descends from the partial Fourier transform $F_{1}$ which leaves (for a suitable normalization) the functions $1_{u (L_{v}^{V})}, 1_{g_{n + 1} (O_{F_{v}})}$ invariant.
Finally, as ${S L}_{2} (F_{v})$ is generated by $(\begin{matrix} 1 p_{F_{v}}^{- 1} \\ 0 \\ 1 \end{matrix})$ and $w$ , we infer that $Φ$ is fixed by ${S L}_{2} (F_{v})$ from which it is relatively straightforward to deduce $Φ = 0$ .

It is also worth mentioning that in a very interesting work, Jingwei Xiao [51] has shown that the Jacquet-Rallis fundamental lemma implies the (usual) endocospic fundamental lemma for unitary groups. Thus, combining his argument with the proof outlined above yields a completely elementary proof of the Langlands-Shelstad fundamental lemma for unitary groups!

The two previous results on smooth transfer and the fundamental lemma are already enough to imply the Gan-Gross-Prasad Conjecture 2.1 under some local restrictions on the cuspidal representation

π

(originating from the use of simple versions of the Jacquet-Rallis trace formulae, allowing to bypass all convergence issues) as was done by W. Zhang in [57]. However, to derive the refinement of Conjecture 2.2 following the same strategy, we need an extra local ingredient relating the local periods of Ichino-Ikeda to similar local distributions associated to the Rankin-Selberg and Flicker-Rallis periods. More precisely, by the work of Jacquet-Piatetskii-Shapiro-Shalika, on the one hand, and Flicker-Rallis, on the other hand,
it is known that the two automorphic periods

P_{H_{1}}

and

P_{H_{2}, η}

admit factorizations of the form

\begin{aligned} (3.1) & P_{H_{1}} (φ) & = \prod_{v}^{'} P_{H_{1}, v} (W_{φ, v}) \\ (3.2) & P_{H_{2}, η} (φ) & = \frac{1}{4} \prod_{v}^{'} P_{H_{2}, η, v} (W_{φ, v}) \end{aligned}

for

φ

a factorizable vector in a given cuspidal automorphic representation

Π = Π_{n + 1} \otimes Π_{n}

G^{'} (A_{F})

, where

W_{φ} (g) = \int_{[N^{'}]} φ (u g) ψ^{'} (u)^{- 1} d u = \prod_{v} W_{φ, v} (g_{v})

denotes a factorization of the Whittaker function of

φ

(here

N^{'}

stands for the standard maximal unipotent subgroup of

G^{'}

and

ψ^{'}

is a nondegenerate character of

[N^{'}]), P_{H_{1}, v}, P_{H_{2}, η, v}

are explicit linear forms on the local Whittaker model

W (Π_{v}, ψ_{v}^{'})

Π_{v}

and the products in (3.1), (3.2) are to be regularized and understood "in the sense of

L

-functions" in a way similar to (2.4).

Based on the factorizations (3.1) and (3.2), the contribution of

Π

to the spectral expansion (2.8) can be shown to itself admit a factorization roughly as the product of local distributions (called relative characters)

I_{Π_{v}}

defined by

I_{Π_{v}} (f_{v}^{'}) = \sum_{W_{v} \in W (Π_{v}, ψ_{v}^{'})} P_{H_{1}, v} (Π_{v} (f_{v}^{'}) W_{v}) \bar{P_{H_{2}, η, v} (W_{v})}, f_{v}^{'} \in C_{c}^{\infty} (G_{v}^{'})

where the sum runs over a suitable orthonormal basis of the Whittaker model. On the other hand, from the Ichino-Ikeda Conjecture 2.2, we expect the contribution of

π ↪ A_{cusp} (G)

to the spectral expansion of (2.6) to essentially factorize into the product of the local relative characters (where again the sum is taken over an orthonormal basis)

J_{π_{v}} (f_{v}) = \sum_{φ_{v} \in π_{v}} P_{H, v} (π_{v} (f_{v}) φ_{v}, φ_{v}), f_{v} \in C_{c}^{\infty} (G_{v})

In [56], W. Zhang has conjectured that the local Jacquet-Rallis transfer

f_{v} \leftrightarrow f_{v}^{'}

also satisfies certain precise spectral relations involving the above relative characters. This is exactly the extra local ingredient needed to finish the proof of the Ichino-Ikeda conjecture based on a comparison of the Jacquet-Rallis trace formulae. This conjecture was shown in [56] to hold for unramified and supercuspidal representations, and the method was further extended and amplified in [13], allowing to prove the conjecture for all (tempered) representations at non-Archimedean places. Later, in [15] I gave a better proof of this conjecture which also has the advantage of working uniformly at all places (including Archimedean ones). To state the result, we introduce some terminology/notation: for a place

v

F

and a smooth irreducible representation

π_{v}

G_{v}

, we denote by

π_{E, v}

the local base-change of

π_{v}

, that is, the smooth irreducible representation of

G_{v}^{'}

whose

L

-parameter is given by composing that of

π_{v}

with the natural embedding of

L

-groups

^{L} G_{v} \to^{L} G_{v}^{'}

, and, moreover, we say that

π_{v}

H_{v}

-distinguished if

{Hom}_{H_{v}} (π_{v}, C) \neq 0

, that is, with the notation of Section 1.1, if the multiplicity

m (π_{v})

equals 1 .

Theorem 3.2. There exist explicit local constants

{(κ_{v})}_{v}

indexed by the set of all places of

F

and satisfying the product formula

\prod_{v} κ_{v} = 1

such that the following property is verified: for every place

v

, every tempered representation

π_{v}

G_{v}

which is

H_{v}

-distinguished and every pair

(f_{v}, f_{v}^{'}) \in C_{c}^{\infty} (G_{v}) \times C_{c}^{\infty} (G_{v}^{'})

of matching functions (that is,

f_{v} \leftrightarrow f_{v}^{'}

), we have

\begin{matrix} (3.3) & I_{π_{v, E}} (f_{v}^{'}) = κ_{v} J_{π_{v}} (f_{v}) \end{matrix}

Moreover, the above identities characterize the Jacquet-Rallis transfer, that is, if two functions

f_{v} \in C_{c}^{\infty} (G_{v}), f_{v}^{'} \in C_{c}^{\infty} (G_{v}^{'})

satisfy (3.3) for every tempered irreducible representation

π_{v}

G_{v}

that is

H_{v}

-distinguished, then these functions are transfers of each other.

The proof given in [15] of the above theorem is mainly based on another ingredient of independent interest which is an explicit Plancherel decomposition for the space

G_{v}^{'} / H_{2, v}

or rather, decomposing this quotient as a product in a natural way, for the symmetric variety

{G L}_{n} (E_{v}) / {G L}_{n} (F_{v})

. This spectral decomposition is roughly obtained by applying the Plancherel formula for the group

{G L}_{n} (E_{v})

to a family of zeta integrals, depending on a complex parameter

s

, introduced by Flicker and Rallis [22] and that represents local Asai

L

-factors and taking the residue at

s = 1

of the resulting expression. We will not describe the exact process here, but just mention that this settles in the case at hand a general conjecture of Sakellaridis-Venkatesh [41] on the spectral decomposition of spherical varieties. This Plancherel formula is then used to write the explicit spectral expansion for a local analog of the Jacquet-Rallis trace formula (2.8) which is then compared with a local counterpart of the trace formula (2.6) yielding as a consequence Theorem 3.2 above. Moreover, as another byproduct of this local comparison, we also get a formula conjectured by Hiraga-Ichino-Ikeda for the formal degree of discrete series [29] in the case of unitary groups.

4. GLOBAL ANALYSIS OF JACQUET-RALLIS TRACE FORMULAE

With all the local ingredients explained in the previous section in place, the only remaining tasks to finish the program initiated by Jacquet and Rallis to prove the GanGross-Prasad and Ichino-Ikeda conjectures are global. More specifically, although simple versions of the Jacquet-Rallis trace formulae have been successfully used to establish these conjectures under some local restrictions [13,57], in order to detect all the relevant cuspidal representations of unitary groups, we need more refined versions of the geometric and spectral expansions of (2.6) and (2.8).

As a first important step in that direction, Zydor [58,59] has completely regularized the singular contributions to the geometric sides. We can summarize his main results as follows: for all test functions

f \in C_{c}^{\infty} (G (A_{F}))

and

f^{'} \in C_{c}^{\infty} (G^{'} (A_{F}))

, there exist "canonical" regularization of the (usually divergent) integrals (2.5) and (2.7), that we will still denote by

J (f)

and

I (f^{'})

, as well as decompositions

\begin{matrix} (4.1) & J (f) = \sum_{δ \in (H ∖ G / / H) (F)} O (δ, f) and I (f^{'}) = \sum_{γ \in (H_{1} ∖ G^{'} / / H_{2}) (F)} O_{η} (γ, f^{'}) \end{matrix}

where

H ∖ G / / H

and

H_{1} ∖ G^{'} / / H_{2}

stand for the corresponding categorical quotients and

O (δ, \cdot), O_{η} (γ, \cdot)

are distributions supported on the union of the adelic double cosets with images

δ

and

γ

(H ∖ G / / H) (A_{F})

and

(H_{1} ∖ G^{'} / / H_{2}) (A_{F})

, respectively, which coincide with the previously defined relative orbital integrals when

δ

and

γ

are regular semisimple.

Zydor obtains these regularized orbital integrals by adapting a truncation procedure developed by Arthur in the context of the usual trace formula to the relative setting at hand. It should be emphasized that contrary to what happens with Arthur's trace formula, the resulting distributions are directly invariant (in a relative sense, that is, here under the natural action of

H \times H

H_{1} \times H_{2}

) and do not depend on any auxiliary choice (such as that of a maximal compact subgroup). It is in this sense that the regularizations of Zydor are really "canonical." It should be mentioned that another, different, approach to such regularization was proposed by Sakellaridis [40] in the context of general relative trace formulae. It is based on analyzing the exponents at infinity of generalized theta series together with a natural procedure to regularize integrals of multiplicative functions when the corresponding character is nontrivial.

Before we even consider the analogous, more subtle, regularization problem on the spectral side, there appears the natural question of how to compare the singular contributions to the refined geometric expansions of (4.1). This issue was completely resolved in a very long paper [20] by Chaudouard and Zydor. To state their main result, it is convenient to again consider the relevant pure inner forms of

G

(as defined in Section 1.1): for every Hermitian space

W^{'}

of the same dimension as

W

, we have a relevant pure inner form

G^{W^{'}} =

U (V^{'}) \times U (W^{'})

with its diagonal subgroup

H^{W^{'}} = U (W^{'})

where

V^{'} = W^{'} \oplus^{⊥} W^{⊥}

. Moreover, the correspondence of orbits (2.9) extends to an isomorphism between categorical quotients,

\begin{matrix} (4.2) & H ∖ G / / H ≃ H_{1} ∖ G^{'} / / H_{2} \end{matrix}

and for every

W^{'}

as before,

H^{W^{'}} ∖ G^{W^{'}} / / H^{W^{'}}

can naturally be identified with

H ∖ G / / H

. With these preliminaries, the main result of Chaudouard and Zydor can now be stated as follows:

Theorem 4.1 (Chaudouard-Zydor). Assume that

f^{W^{'}} = \prod_{v} f_{v}^{W^{'}} \in C_{c}^{\infty} (G^{W^{'}} (A_{F}))

, where

W^{'}

runs over all isomorphism classes of Hermitian spaces of dimension

n

, and

f^{'} = \prod_{v} f_{v}^{'} \in C_{c}^{\infty} (G^{'} (A_{F}))

are factorizable test functions such that for every place

v

, and each

W^{'}, f_{v}^{W^{'}}

and

f_{v}^{'}

are Jacquet-Rallis transfers of each other (that is,

f_{v}^{W^{'}} \leftrightarrow f_{v}^{'}

). Then, for every

δ \in (H ∖ G / / H) (F)

with image

γ \in (H_{1} ∖ G^{'} / / H_{2}) (F)

by (4.2), we have

\begin{matrix} (4.3) & \sum_{W^{'}} O (δ, f^{W^{'}}) = O_{η} (γ, f^{'}) \end{matrix}

It should be noted that when

δ

, hence also

γ

, is regular semisimple, the left-hand sum in (4.3) only contains one nonidentically vanishing term but that in general more than one relevant pure inner forms can contribute. Also, the above result extends to nonfactorizable test functions, provided the wording is changed suitably.

The next natural step would be to develop regularized spectral expansions similar to (4.1). As a first result in that direction, Zydor has shown decompositions of the form

\begin{matrix} (4.4) & J (f) = \sum_{χ \in X (G)} J_{χ} (f) and I (f^{'}) = \sum_{χ^{'} \in X (G^{'})} I_{χ^{'}} (f^{'}) \end{matrix}

where

X (G)

and

X (G^{'})

stand for the set of cuspidal data of the groups

G

and

G^{'}

respectively, that is the sets of pairs

(M, σ)

where

M

is a Levi subgroup (of

G

G^{'}

) and

σ

is a cuspidal automorphic representation of

M (A_{F})

taken up to conjugacy (by

G (F)

G^{'} (F)

). According to Langlands theory of pseudo-Eisenstein series, these sets index natural equivariant Hilbertian decompositions:

L^{2} ([G]) = ⨁_{χ \in X (G)} L_{χ}^{2} ([G]), L^{2} ([G^{'}]) = ⨁_{χ^{'} \in X (G^{'})} L_{χ^{'}}^{2} ([G^{'}])

The automorphic kernels

K_{f}, K_{f^{'}}

decompose accordingly into series

K_{f} = \sum_{χ} K_{f, χ}

K_{f^{'}} = \sum_{χ^{'}} K_{f^{'}, χ^{'}}

where

K_{f, χ}

and

K_{f^{'}, χ}

are kernel functions representing the restrictions

R_{χ} (f)

and

R_{χ^{'}} (f^{'})

of the right convolution operators

R (f)

and

R (f^{'})

L_{χ}^{2} ([G])

and

L_{χ^{'}}^{2} ([G^{'}])

, respectively. The distributions

f \mapsto J_{χ} (f)

and

f^{'} \mapsto I_{χ^{'}} (f^{'})

are then roughly defined by applying the same regularization procedure that Zydor used for the expressions

J (f)

and

I (f^{'})

up to replacing the integrands by

K_{f, χ}

and

K_{f^{'}, χ^{'}}

respectively, that is, in symbolic terms:

\begin{aligned} (4.5) & J_{χ} (f) & = \int_{[H] \times [H]}^{r e g} K_{f, χ} (h_{1}, h_{2}) d h_{1} d h_{2} \\ I_{χ^{'}} (f^{'}) & = \int_{[H_{1}] \times [H_{2}]}^{r e g} K_{f^{'}, χ^{'}} (h_{1}, h_{2}) η (h_{2}) d h_{1} d h_{2} \end{aligned}

However, the expansions (4.4) are of little use as they stand and need to be suitably refined to allow for a meaningful comparison of the trace formulae. In Arthur's terminology, (4.4) are coarse spectral expansions and we need refined spectral expansions for each of the terms

J_{χ} (f)

I_{χ^{'}} (f^{'})

This problem has so far proved to be a very difficult for general cuspidal data

χ

and

χ^{'}

. However, a recent result of mine in collaboration with Y. Liu, W. Zhang, and X. Zhu [17] allows isolating in the coarse spectral expansions (4.4) the only terms that are eventually of interest consequently reducing the problem to some very particular cuspidal data

χ^{'}

G^{'}

The result proved in [17] is very general so let us place ourself for one moment in the framework of an arbitrary connected reductive group

G

over the number field

F

. Let

Σ

be a set of non-Archimedean places of

F

(possibly infinite) such that for each

v \in

Σ

, the group

G_{v}

is unramified and fix a hyperspecial compact subgroup

K_{v} \subset G_{v}

with

K_{v} = G (O_{v})

for almost all

v \in Σ

. We let

X_{Σ} (G)

be the set of

Σ

-unramified cuspidal data of

G

, that is, the cuspidal data represented by pairs

(M, σ)

with

σ

unramified at all places of

v \in Σ

(with respect to

K_{v}

or, rather, the hyperspecial subgroup it induces in

M_{v}

). For

χ \in X_{Σ} (G)

, we define its

Σ

-near equivalence class, henceforth denoted by

N_{Σ} (χ)

, as the set of all cuspidal data

χ^{'} \in X_{Σ} (G)

such that if

χ

and

χ^{'}

are represented by pairs

(M, σ)

and

(M^{'}, σ^{'})

respectively, then there exist automorphic unramified characters

λ

and

λ^{'}

M (A_{F})

and

M^{'} (A_{F})

, respectively, with the property that for every

v \in Σ

the Satake parameters of the unique

K_{v}

-unramified subquotients in

I_{P_{v}}^{G_{v}} (σ_{v} \otimes λ_{v})

and

I_{P_{v}^{'}}^{G_{v}} (σ_{v}^{'} \otimes λ_{v}^{'}) (

where

P, P^{'}

are arbitrary chosen parabolics with Levi components

M, M^{'}

) are isomorphic. We also fix a compact-open subgroup

K = \prod_{v \in S_{f}} K_{v}

G (A_{f})

(where

S_{f}

denotes the set of finite places of

F

and

K_{v}

coincides with the previous choice of hyperspecial subgroup when

v \in Σ

) and we define the Schwartz space of

K

-biinvariant functions on

G (A_{F})

as the restricted tensor product

ς_{K} (G (A_{F})) = S (G (F_{\infty})) \otimes ⨂_{v \in S_{f}}^{'} C_{c} (K_{v} ∖ G_{v} / K_{v})

where

C_{c} (K_{v} ∖ G_{v} / K_{v})

denotes the space of bi-

K_{v}

-invariant compactly supported functions on

G_{v}

(that is the

K_{v}

-spherical Hecke algebra when

v \in Σ

F_{\infty}

is the product of the Archimedean completions of

F

and

S (G (F_{\infty}))

stands for the Schwartz space of the reductive Lie group

G (F_{\infty})

in the sense of [19]. More precisely,

S (G (F_{\infty}))

is the space of smooth functions

f : G (F_{\infty}) \to C

such that for every polynomial differential operator on

G (F_{\infty})

, the derivatives

D f

is bounded or, equivalently, such that for every left- (or right)invariant differential operator

X, X f

is decreasing faster than the inverse of any polynomial on

G (F_{\infty})

The Schwartz space

S (G (F_{\infty}))

is naturally a FrÃ©chet algebra under the convolution product and we also set

M_{\infty} (G) = {End}_{cont, s (G (F_{\infty})) - bimod} (S (G (F_{\infty})))

for the space of continuous endomorphisms of

S (G (F_{\infty}))

seen as a bimodule over itself. This is an algebra acting on any smooth admissible FrÃ©chet representation of moderate growth of

G (F_{\infty})

in the sense of Casselman-Wallach. Moreover, as an application of a form of Schur lemma, for every irreducible Casselman-Wallach representation

π_{\infty}

G (F_{\infty})

and every

μ_{\infty} \in M_{\infty} (G)

there exists a scalar

μ_{\infty} (π_{\infty}) \in C

such that

π_{\infty} (μ_{\infty}) = μ_{\infty} (π_{\infty}) I d

. Thus,

M_{\infty} (G)

can be seen as some big algebra of multipliers for

δ (G (F_{\infty}))

. We also define the algebra of

Σ

-multipliers as the restricted tensor product

M_{Σ} (G) = M_{\infty} (G) ⨂_{v \in Σ}^{'} H (G_{v}, K_{v})

where, for

v \in Σ, H (G_{v}, K_{v}) = C_{c} (K_{v} ∖ G_{v} / K_{v})

is the spherical Hecke algebra. Then,

M_{Σ} (G)

acts naturally on the global Schwartz space

ς_{K} (G (A_{F}))

, and we shall denote this action as the convolution product

*

. One of the main result of [17] can now be stated as follows:

Theorem 4.2 (Beuzart-Plessis-Liu-Zhang-Zhu). Let

χ \in X_{Σ} (G)

. Then, there exists a multiplier

μ_{χ} \in M_{Σ} (G)

such that for every Schwartz function

f \in S_{K} (G (A_{F}))

and every other cuspidal datum

χ^{'} \in X_{Σ} (G)

, we have

R_{χ^{'}} (μ_{χ} * f) = {\begin{cases} R_{χ^{'}} (f) & if χ^{'} \in N_{Σ} (χ) \\ 0 & otherwise \end{cases}

The above theorem can be roughly paraphrased by saying that the multiplier

μ_{χ}

"isolates" the near-equivalence class

N_{Σ} (χ)

from the other cuspidal data. A large part of the proof given in [17] consists in establishing the existence of a large subalgebra of

M_{\infty} (G)

which admits an explicit spectral description, that is, through its action on irreducible Casselman-Wallach representations of

G (F_{\infty})

. The algebra thus constructed generalizes Arthur's multipliers [5] and, moreover, builds on previous work of Delorme [21].

Going back to the setting of the Jacquet-Rallis trace formulae, the above theorem can be applied to isolate in the expansions (4.4) the automorphic

L

-packet of a given cuspidal automorphic representation

π

G (A_{F})

, on the one hand, and the cuspidal datum

χ

G^{'}

"supporting" its base-change

π_{E}

, on the other hand. Moreover, essentially using the spectral characterization of Theorem 3.2 for the transfer, this can be done by multipliers

μ_{π} \in M_{Σ} (G)

and

μ_{χ} \in M_{Σ} (G^{'})

that are compatible with the Jacquet-Rallis transfer in the following sense: if

f = \prod_{v} f_{v} \in S_{K} (G (A_{F}))

and

f^{'} = \prod_{v} f_{v}^{'} \in S_{K^{'}} (G^{'} (A_{F}))

are transfers of each other then so are

μ_{π} * f

and

μ_{χ} * f^{'}

(where here we take

Σ

to consist of almost all places that split in

E

and for

K, K^{'}

arbitrary compact-open subgroups of

G (A_{f}), G^{'} (A_{f})

that are hyperspecial at places in

Σ

). All in all, applying these multipliers to global test functions

f

and

f^{'}

that are transfers of each other, and comparing the geometric expansions (4.1), we obtain an identity of the following shape:

\sum_{W^{'}} \sum_{\begin{matrix} π^{'} ↪ A_{cusp} (G^{W^{'}}) \\ π_{E}^{'} = π_{E} \end{matrix}} J_{π^{'}} (f) = I_{χ} (f^{'})

where the outside left sum runs over isomorphism classes of Hermitian spaces of the same dimension as

W

(or, equivalently, relevant pure inner forms of

G

). Besides, as a consequence of the local Gan-Gross-Prasad conjecture, when

π_{E}

is generic, the left-hand side contains at most one nonzero term. Thus, as a final step to establish the Gan-Gross-Prasad and IchinoIkeda conjectures, it only remains to analyze the distribution

I_{χ}

. When the base-change

π_{E}

is itself cuspidal, that is, when

χ = {(G^{'}, π_{E})}

, by the works of Jacquet-PiatetskiShapiro-Shalika and Flicker-Rallis already recalled,

I_{χ}

essentially factors as the product of the local relative characters

I_{π_{E, v}}

and Theorem 3.2 then allows to conclude. However, in general a similar factorization of

I_{χ}

is far from obvious and was actually established in my joint work with Chaudouard and Zydor [16]. It is exactly of the shape predicted by the Ichino-Ikeda conjecture. More precisely:

Theorem 4.3 (Beuzart-Plessis-Chaudouard-Zydor). Let

π

be a cuspidal automorphic representation of

G (A_{F})

whose base-change

π_{E}

is generic. Let

χ

be the cuspidal datum of

G^{'}

such that

π_{E}

contributes to the spectral decomposition of

L_{χ}^{2} ([G^{'}])

. Then, for every factor izable test function

f^{'} = \prod_{v} f_{v}^{'} \in S (G^{'} (A_{F}))

, we have

\begin{matrix} (4.6) & I_{χ} (f^{'}) = \frac{1}{| S_{π} |} \prod_{v}^{'} I_{π_{E, v}} (f_{v}^{'}) \end{matrix}

where the product has to be understood, as for (2.4), "in the sense of L-functions."

In [16], two proofs are actually given of the above theorem: one using truncations operators and the other one based on the global theory of Zeta integrals. For both methods, a crucial step is to spectrally expand the restriction of the Flicker-Rallis period (that is, the integral over

[H_{2}]

) to functions

φ \in L_{χ}^{2} ([G^{'}])

that are sufficiently rapidly decreasing. A consequence of this computation is that this period only depends on the

π_{E}

-component of

φ

and it is mainly for this reason that the contribution of

χ

to the Jacquet-Rallis trace formula

I (f^{'})

is eventually discrete (although in the case at hand,

L_{χ}^{2} ([G^{'}])

usually has a purely continuous spectrum). For this, the truncation method is based on the work of Jacquet-Lapid-Rogawski who have defined and studied generalizations of Arthur's truncation operator to the setting of Galois periods and proved analogs of the Maass-Selberg relations in this context. On the other hand, the other method starts by expressing the Flicker-Rallis period as a residue of the integral over

[H_{2}]

φ

against an Eisenstein series. Unfolding carefully this expression as in the work of Flicker-Rallis, we can rewrite it as a Zeta integral of the sort that represents Asai

L

-functions. The precise location of the poles of these

L

-functions, as well as an explicit residue computation of a family of distributions, then allows to conclude.

Finally, let me mention that in work in progress with P.-H. Chaudouard, we are able to analyze the contributions to the Jacquet-Rallis trace formula of more general cuspidal data

χ \in X (G^{'})

than that appearing in Theorem [16]. The final result is similar to (4.6) except that the right-hand side has to be integrated over a certain family of automorphic representations

π

G (A_{F})

. More precisely, our results include some cuspidal data supporting the basechanges of automorphic representations of

G = U (V) \times U (W)

that are Eisenstein in the first factor and cuspidal in the second. In this particular case, the contribution of the corresponding cuspidal datum to the trace formula

J (f)

is absolutely convergent and a refined spectral expansion can readily be obtained as an integral of Gan-Gross-Prasad periods between a cusp form and an Eisenstein series. These last periods are related, by some unfolding, to Bessel periods of cusp forms on smaller unitary groups. For this reason, our extension of Theorem 4.3 with Chaudouard should have similar applications to the Gan-Gross-Prasad and Ichino-Ikeda conjectures for general Bessel periods.

5. LOOKING FORWARD

As illustrated in the previous sections, various trace formula approaches to the GanGross-Prasad conjectures for unitary groups have been very successful. However, despite these favorable and definite results, these developments also raise interesting questions or have lead to fertile new research direction:

First, there is the question of whether similar techniques can be applied to prove the global Gan-Gross-Prasad conjectures for other groups. Indeed, the original conjectures in [23] also include general Bessel periods on (product of) orthogonal groups $SO (n) \times SO (m) (n ≢ m [2])$ , as well as so-called Fourier-Jacobi periods on unitary groups $U (n) \times U (m) (n \equiv m$ [2]) or symplectic/metaplectic groups $Mp (n) \times Sp (m)$ . In the case of $U (n) \times U (n)$ , a relative trace formula approach
has been proposed by Y. Liu and further developed by H. Xue [52]. However, the situation is not as complete as for the Jacquet-Rallis trace formulae in the case of $U (n + 1) \times U (n)$ . It would be interesting to see if the latest developments, in particular those from my joint work with Chaudouard and Zydor [16], can be adapted to this setting. This could possibly lead to a proof of the GanGross-Prasad conjecture for general Fourier-Jacobi periods on unitary groups. The situation for orthogonal and symplectic/metaplectic groups is much less satisfactory and there is no clear approach through a comparison of relative trace formulae, yet. This is due in particular to the fact that, instead of the FlickerRallis periods, in these cases we are naturally lead to consider period integrals originally studied by Bump-Ginzburg that detect cuspidal automorphic representations of $G L (n)$ of orthogonal type. These period integrals involve the product of two exceptional theta series on a double cover of $GL (n)$ and do not have any obvious geometric realizations (except when $n = 2$ ). This makes the task of writing a geometric expansion for the corresponding trace formulae quite unclear. It would certainly be interesting to see if the recent Hamiltonian duality picture of Ben Zvi-Sakellaridis-Venkatesh can shed some light on this matter (in particular, by associating a Hamiltonian space to the Bump-Ginzburg periods).
In the local setting, the new trace formulae first discovered by Waldspurger [47] and further developed in [12] seem to be of quite broad applicability to all kind of distinction problems. Actually, similar trace formulae have already been established in other contexts $[11, 18, 50]$ with new applications in the spirit of "relative Langlands functorialities" each time. However, all these developments have been made on a case-by-case basis so far and it would be very interesting and instructive to elaborate a general theory. In particular, in view of the proposal by Sakellaridis-Venkatesh [41] of a general framework for the relative Langlands program, we could hope to establish general local relative trace formulae for the $L^{2}$ spaces of spherical varieties $X$ and relate those to the dual group construction of Sakellaridis-Venkatesh.
Finally, in a slightly different direction the general isolation Theorem 4.2 clearly has the potential to be applied in other context, e.g., it would be interesting to see if it can be used as a technical device to simplify some other known comparison of trace formulae. Another intriguing question is to look for a precise spectral description of the (abstract) multiplier algebra $M_{\infty} (G)$ and in [17], we actually argue that $M_{\infty} (G)$ should be seen as the natural Archimedean analog of the Bernstein center for $p$ -adic groups.

ACKNOWLEDGMENTS

First and foremost, I thank my former PhD advisor Jean-Loup Waldspurger for introducing me to this fertile circle of ideas and for always being so generous in sharing his insights on these problems.

I am also grateful to my colleagues and coauthors Pierre-Henri Chaudouard, Patrick Delorme, Wee-Teck Gan, Yifeng Liu, Dipendra Prasad, Yiannis Sakellaridis, Chen Wan, Wei Zhang, Xinwen Zhu, and Michal Zydor for many inspiring discussions and exchanges over the years.

FUNDING

The author's work was partially supported by the Excellence Initiative of Aix-Marseille University (A*MIDEX), a French "Investissements d'Avenir" program.

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RAPHAÃ‹L BEUZART-PLESSIS

Aix Marseille Univ, CNRS, I2M, Marseille, France, raphael.beuzart-plessis@univ-amu.fr

THE COHOMOLOGY OF SHIMURA VARIETIES WITH TORSION COEFFICIENTS

ANA CARAIANI

ABSTRACT

In this article, we survey recent work on some vanishing conjectures for the cohomology of Shimura varieties with torsion coefficients, under both local and global conditions. We discuss the

p

-adic geometry of Shimura varieties and of the associated Hodge-Tate period morphism, and explain how this can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato-Tate conjecture for elliptic curves over CM fields.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G18; Secondary 11R39, 11F33, 11F70, 11F75, 11F80

KEYWORDS

Shimura varieties, locally symmetric spaces, automorphic representations, Galois representations,

p

-adic Langlands programme

1. INTRODUCTION

Shimura varieties are algebraic varieties defined over number fields and equipped with many symmetries, which often provide a geometric realization of the Langlands correspondence. After base change to

C

, they are closely related to certain locally symmetric spaces, but the beauty of Shimura varieties lies in their rich arithmetic.

To describe a Shimura variety, one needs to start with a Shimura datum

(G, X)

. Here,

G

is a connected reductive group over

Q

and

X

is a conjugacy class of homomorphisms

h : {Res}_{C / R} G_{m} \to G_{R}

of algebraic groups over

R

. Both

G

and

X

are required to satisfy certain highly restrictive axioms, cf. [22, Â§2.1]. In particular, this allows one to give the conjugacy class

X

a more geometric flavor, as a variation of polarisable Hodge structures. One can show that such an

X

is a disjoint union of finitely many copies of Hermitian symmetric domains.

Let

K \subset G (A_{f})

be a sufficiently small compact open subgroup (the precise technical condition is called "neat"). The double quotient

G (Q) ∖ X \times G (A_{f}) / K

, a priori a complex manifold, comes from an algebraic variety

S_{K}

defined over a number field

E

, called the reflex field of the Shimura datum. The varieties

S_{K}

are smooth and quasiprojective. Their Ã©tale cohomology groups (with or without compact support)

H_{(c)}^{*} (S_{K} \times_{E} \bar{Q}, Q_{ℓ})

are equipped with two kinds of symmetries. There is a Hecke symmetry coming from varying the level, i.e., the compact open subgroup

K

, and considering various transition morphisms between Shimura varieties at different levels. There is also a Galois symmetry, coming from the natural action of

Gal (\bar{E} / E)

on Ã©tale cohomology.

For this reason, Shimura varieties have played an important role in realizing instances of the global Langlands correspondence over number fields. Indeed, a famous conjecture of Kottwitz predicts the relationship between the Galois representations occurring in the

ℓ

-adic Ã©tale cohomology of the Shimura varieties for

G

and those Galois representations associated with (regular,

C

-algebraic) cuspidal automorphic representations of

G

. See [64, REMARK 1.1.1] for a modern formulation of this conjecture.

There is a complete classification of groups that admit a Shimura datum. For example, if

G = {G S p}_{2 n}

, one can take

X

to be the Siegel double space

\begin{matrix} (1.1) & {Z \in M_{n} (C) ∣ Z = Z^{t}, Im (Z) positive or negative definite} \end{matrix}

The associated Shimura varieties are called Siegel modular varieties and they are moduli spaces of principally polarized abelian varieties. Many other Shimura varieties - those of socalled "abelian type" - can be studied using moduli-theoretic techniques, by relating them to Siegel modular varieties. See [39] for an excellent introduction to the subject, which is focused on examples.

In this article, we will be primarily concerned with the geometry of the Shimura varieties

S_{K}

, after base change to a

p

-adic field, as well as with their Ã©tale cohomology groups

H_{(c)}^{*} (S_{K} \times_{E} \bar{Q}, F_{ℓ})

with torsion coefficients. These groups are much less understood than their characteristic zero counterparts. We discuss certain conjectures about when these cohomology groups are expected to vanish, under both global and local conditions. Furthermore, we explain how the geometry of the Hodge-Tate period morphism, introduced in [53] and
refined in [17], can be used to make progress on these conjectures. Finally, we describe some applications of these results, in particular to the proof of the Sato-Tate conjecture for elliptic curves over CM fields [1].

2. A VANISHING CONJECTURE FOR LOCALLY SYMMETRIC SPACES

Let

G / Q

be a connected reductive group. We consider the symmetric space associated with the Lie group

G (R)

, which we define as

X = G (R) / K_{\infty}^{\circ} A_{\infty}^{\circ}

. Here,

K_{\infty}^{\circ}

is the connected component of the identity in a maximal compact subgroup

K_{\infty} \subset G (R)

, and

A_{\infty}^{\circ}

is the connected component of the identity inside the real points of the maximal

Q

-split torus in the center of

G

. Given a neat compact open subgroup

K \subset G (A_{f})

, we can form the double quotient

X_{K} = G (Q) ∖ X \times G (A_{f}) / K

, which we call a locally symmetric space for

G

. This is a smooth Riemannian manifold, which does not have a complex structure, in general.

Example 2.1. If

G = {S L}_{2} / Q

, we can identify

X = {S L}_{2} (R) / {S O}_{2} (R)

with the upper halfplane

H^{2} = {z \in C ∣ Im z > 0}

equipped with the hyperbolic metric, on which

{S L}_{2} (R)

acts transitively by the isometries

z \mapsto \frac{a z + b}{c z + d} for z \in H^{2} and (\begin{array}{ll} a & b \\ c & d \end{array}) \in {S L}_{2} (R)

Under this action,

{S O}_{2} (R)

is the stabilizer of the point

i

. By strong approximation for

{S L}_{2} / Q

, for any compact open subgroup

K \subseteq {S L}_{2} (\hat{Z})

, there is only one double coset

{S L}_{2} (Q) ∖ {S L}_{2} (A_{f}) / K

. Write

Γ = {S L}_{2} (Q) \cap K

, which will be a congruence subgroup contained in

{S L}_{2} (Z)

. The locally symmetric spaces

X_{K}

can be identified with quotients

Γ ∖ H^{2}

. For

Γ

neat, these quotients inherit the complex structure on

H^{2}

and can be viewed as Riemann surfaces. Even more, these quotients arise from algebraic curves called modular curves, which are defined over finite extensions of

Q

. Modular curves are examples of (connected) Shimura varieties. They represent moduli problems of elliptic curves endowed with additional structures. Even though they are some of the simplest Shimura varieties (the main complication being that they are noncompact), their geometry is already fascinating.

However, let

F / Q

be an imaginary quadratic field and take

G = {Res}_{F / Q} {S L}_{2}

. Then we can identify the symmetric space

X = {S L}_{2} (C) / {S U}_{2} (R)

with 3-dimensional hyperbolic space

H^{3}

. Once again, we can identify the locally symmetric spaces

X_{K}

with quotients

Γ ∖ H^{3}

, where

Γ = {S L}_{2} (F) \cap K

is a congruence subgroup. In this case, the locally symmetric spaces are arithmetic hyperbolic 3-manifolds and do not admit a complex structure. In particular, we cannot speak of Shimura varieties in this setting.

In general, Shimura varieties are closely related to locally symmetric spaces, as in the first example, though the latter are much more general objects. For example, the locally symmetric spaces for

G = {Res}_{F / Q} {G L}_{n}

do not arise from Shimura varieties if

n \geq 3

, and, for

n = 2

, they can only be related to Shimura varieties if

F

is a totally real field. In some instances, such as for

{Res}_{F / Q} {G L}_{2}

with

F

a totally real field, one needs to replace

G (R) / K_{\infty}^{\circ} A_{\infty}^{\circ}

by a slightly different quotient in order to obtain Shimura varieties.

^{1}

We now define the invariants

l_{0} = rk (G (R)) - rk (K_{\infty}) - rk (A_{\infty}) and q_{0} = \frac{1}{2} (\dim_{R} (X) - l_{0})

These were first introduced by Borel-Wallach in [5]. There, they show up naturally in the computation of the (

g, K_{\infty}

)-cohomology of tempered representations of

G (R)

. In the Shimura variety setting, we consider the variants

l_{0} = l_{0} (G^{ad})

and

q_{0} = q_{0} (G^{ad})

because of the different quotient used. In this case,

l_{0} (G^{ad})

can be shown to be equal to 0 by the second axiom in the definition of a Shimura datum.

K

varies, we have a tower of locally symmetric spaces

{(X_{K})}_{K}

, on which a spherical Hecke algebra

T

for

G

acts by correspondences. The systems of Hecke eigenvalues occurring in the cohomology groups

H_{(c)}^{*} (X_{K}, C)

or, equivalently, the maximal ideals of

T

in the support of these cohomology groups, can be related to automorphic representations of

G (A_{f})

by work of Franke and Matsushima [29]. The goal of this section is to state a conjecture on the cohomology of locally symmetric spaces with torsion coefficients

F_{ℓ}

, where

ℓ

is a prime number. This conjecture is formulated in [25] (see the discussion around Conjecture 3.3) and in [12, CONJECTURE B]. Roughly, it says that the part of the cohomology outside the range of degrees

[q_{0}, q_{0} + l_{0}]

is somehow degenerate. Note that this range of degrees is symmetric about the middle

\frac{1}{2} \dim_{R} X

of the total range of cohomology and, in the Shimura variety case, it equals the middle degree of cohomology.

To formulate this more precisely, we use the notion of a non-Eisenstein maximal ideal in the Hecke algebra, for which we need to pass to the Galois side of the global Langlands correspondence. For simplicity, we will restrict our formulation to the case of

G =

{Res}_{F / Q} {G L}_{n}

for some number field

F

, although the conjecture makes sense more generally. Let

T

be the abstract spherical Hecke algebra away from a finite set

S

of primes of

F

and let

m \subset T

be a maximal ideal in the support of

H_{(c)}^{*} (X_{K}, F_{ℓ})

. Assume that there exists a continuous, semisimple Galois representation

{\bar{ρ}}_{m} : Gal (\bar{F} / F) \to {G L}_{n} ({\bar{F}}_{ℓ})

associated with

m

: by this, we mean that

{\bar{ρ}}_{m}

is unramified at all the primes of

F

away from the finite set

S

, and that, at any prime away from

S

, the Satake parameters of

m

match the Frobenius eigenvalues of

{\bar{ρ}}_{m}

. (The precise condition is in terms of the characteristic polynomial of

{\bar{ρ}}_{m}

applied to the Frobenius at such a prime and depends on various choices of normalizations. See, for example, [1, THEOREM 2.3.5] for a precise formulation.) Since the Galois representation is assumed to be semisimple and we are working with

{Res}_{F / Q} {G L}_{n}

, this property will characterize

{\bar{ρ}}_{m}

by the Cebotarev density theorem and the Brauer-Nesbitt theorem. We say that

m

is non-Eisenstein if such a

{\bar{ρ}}_{m}

is absolutely irreducible.

The existence of

{\bar{ρ}}_{m}

as above should be thought of as a

mod ℓ

version of the global Langlands correspondence, in the automorphic-to-Galois direction; in the case

F = Q

, this existence was conjectured by Ash [4]. The striking part of this conjecture is that it should apply to torsion classes in the cohomology of locally symmetric spaces, not just to those classes that lift to characteristic zero, and which can be related to automorphic representations of

G

. For general number fields, the existence of such Galois representations seems out of reach at the moment, even for classes in characteristic zero!

However, let

F

be a CM field: using nonstandard terminology, we mean that

F

is either a totally real field or a totally complex quadratic extension thereof. In this case, Scholze constructed such Galois representations in the breakthrough paper [53]. This strengthened previous work [33] that applied to cohomology with

Q_{ℓ}

-coefficients. Both these results relied, in turn, on the construction of Galois representations in the self-dual case, due to many people, including Clozel, Kottwitz, Harris-Taylor [34], Shin [61], and ChenevierHarris [21].

We can now state the promised vanishing conjecture for the cohomology of locally symmetric spaces with

F_{ℓ}

-coefficients.

Conjecture 2.2. Assume that

m \subset T

is a non-Eisenstein maximal ideal in the support of

H_{(c)}^{*} (X_{K}, F_{ℓ})

. Then

H_{(c)}^{i} {(X_{K}, F_{ℓ})}_{m} \neq 0

only if

i \in [q_{0}, q_{0} + l_{0}]

In the two examples discussed in Example 2.1, this conjecture can be verified "by hand," since one only needs to control cohomology in degree 0 (the top degree of cohomology can be controlled using PoincarÃ© duality). In the case of

{G L}_{2} / Q

, one can show that the systems of Hecke eigenvalues

m

in the support of

H^{0} (X_{K}, F_{ℓ})

satisfy

\begin{matrix} (2.1) & {\bar{ρ}}_{m} ≃ χ \oplus χ_{c y c l o} \cdot χ \end{matrix}

where

χ

is a suitable

mod ℓ

character of

Gal (\bar{Q} / Q)

and

χ_{cyclo} : Gal (\bar{Q} / Q) \to F_{ℓ}^{\times}

is the

mod ℓ

cyclotomic character. Later, we will introduce a local genericity condition at an auxiliary prime

p \neq ℓ

and we will see that the

{\bar{ρ}}_{m}

in (2.1) also fail to satisfy genericity everywhere. In addition to these and a few more low-dimensional examples, one can also consider the analogue of Conjecture 2.2 for

H_{(c)}^{*} (X_{K}, Q_{ℓ})

. This analogue is related to Arthur's conjectures on the cohomology of locally symmetric spaces [3] and can be verified for

{G L}_{n}

over

C M

fields using work of Franke and Borel-Wallach (see [1, theOREM 2.4.9]).

Conjecture 2.2 is motivated by the Calegari-Geraghty enhancement [12] of the classical Taylor-Wiles method for proving automorphy lifting theorems. The classical method works well in settings where the (co)homology of locally symmetric spaces is concentrated in one degree, for example, for

{G L}_{2} / Q

after localizing at a non-Eisenstein maximal ideal, or for definite unitary groups over totally real fields. In general, however, a certain numerical coincidence that is used to compare the Galois and automorphic sides breaks down. Calegari and Geraghty had a significant insight: they reinterpret the failure of the numerical coincidence in terms of the invariant

l_{0}

. More precisely,

l_{0}

arises naturally from a computation on the Galois side, and the commutative algebra underlying the method can be adjusted if one knows that the cohomology on the automorphic side, after localizing at a non-Eisenstein maximal ideal, is concentrated in a range of degrees of length at most

l_{0}

. For an overview of the key ideas involved in the Calegari-Geraghty method, see [10, Â§10].

In the case of Shimura varieties, Conjecture 2.2 predicts that the non-Eisenstein part of the cohomology with

F_{ℓ}

-coefficients is concentrated in the middle degree. The initial progress on this conjecture in the Shimura variety setting had rather strong additional assumptions: for example, one needed

ℓ

to be an unramified prime for the Shimura datum and

K_{ℓ}

to be hyperspecial, as in the work of Dimitrov [23] and Lan-Suh [40,41]. The theory of perfectoid Shimura varieties and their associated Hodge-Tate period morphism has been a game-changer in this area. For the rest of this article, we will discuss more recent progress on Conjecture 2.2 and related questions in the special case of Shimura varieties, as well as applications that go beyond the setting of Shimura varieties.

3. THE HODGE-TATE PERIOD MORPHISM

The Hodge-Tate period morphism was introduced by Scholze in his breakthrough paper [53] and it was subsequently refined in [17]. It gives an entirely new way to think about the geometry and cohomology of Shimura varieties. In the past decade, it had numerous striking applications to the Langlands programme: to Scholze's construction of Galois representations for torsion classes, to the vanishing theorems discussed in Sections 4 and 5, to the construction of higher Coleman theory by Boxer and Pilloni [8], and to a radically new approach to the Fontaine-Mazur conjecture due to Pan [48].

For simplicity, let us consider a Shimura datum

(G, X)

of of Hodge type. By this, we mean that

(G, X)

admits a closed embedding into a Siegel datum

(\tilde{G}, \tilde{X})

, where

\tilde{G} = {G S p}_{2 n}

, for some

n \in Z_{\geq 1}

, and

\tilde{X}

is as in (1.1). For example,

(G, X)

could be a Shimura datum of

P E L

type arising from a unitary similitude group: the corresponding Shimura varieties will represent a moduli problem of abelian varieties equipped with extra structures (polarizations, endomorphisms, and level structures). This unitary case will be the main example to keep in mind, as this will also play a central role in Section 4.

For some representative

h \in X

, we consider the Hodge cocharacter

μ = h \times {_{R} C |}_{1 st G_{m} factor} : G_{m, C} \to G_{C}

The axioms in the definition of the Shimura datum imply that

μ

is minuscule. The reflex field

E

is the field of definition of the conjugacy class

{μ}

; it is a finite extension of

Q

and the corresponding Shimura varieties admit canonical models over

E

. The cocharacter

μ

also determines two opposite parabolic subgroups

P_{μ}^{std}

and

P_{μ}

, whose conjugacy classes are defined over

E

. These are given by

P_{μ}^{std} = {g \in G ∣ lim_{t \to \infty} ad (μ (t)) g exists}, P^{μ} = {g \in G ∣ lim_{t \to 0} ad (μ (t)) g exists}

We let

{F l}^{std}

and

F l

denote the associated flag varieties, which are also defined over

E

Here is a more moduli-theoretic way to think about these the two parabolics. The chosen symplectic embedding

(G, X) ↪ (\tilde{G}, \tilde{X})

gives rise to a faithful representation

V

G

. The embedding also gives rise to an abelian scheme

A_{K}

over the Shimura variety

S_{K}

at some level

K = G (A_{f}) \cap \tilde{K}

, obtained by restricting the universal abelian scheme over the Siegel modular variety at level

\tilde{K} \subset \tilde{G} (A_{f})

. The cocharacter

μ

induces a grading of

V_{C}

,
which in turn defines two filtrations on

V_{C}

, a descending one Fil

^{∙}

and an ascending one Fil

\cdot

The parabolic

P_{μ}^{std}

is the stabilizer of Fil

^{∙}

, which is morally the Hodge-de Rham filtration on the Betti cohomology of

A_{K}

. There is a holomorphic,

G (R)

-equivariant embedding

\begin{matrix} (3.1) & π_{d R} : X ↪ {F l}^{s t d} (C) = G (C) / P_{μ}^{s t d} \end{matrix}

called the Borel embedding, defined by

h \mapsto {F i l}^{∙} (μ_{h})

. The axioms of a Shimura datum imply that

X

is a variation of polarisable Hodge structures of abelian varieties. Modulitheoretically,

π_{d R}

sends a Hodge structure, such as

H^{1} (A, Q) \otimes_{Q} C ≃ H^{0} (A, Ω_{A}^{1}) \oplus H^{1} (A, O_{A})

to the associated Hodge-de Rham filtration, e.g.,

H^{0} (A, Ω_{A}^{1}) \subset H^{1} (A, Q) \otimes Q C

. The embedding

π_{d R}

is an example of a period morphism. Historically, it has played an important role in the construction of canonical models of automorphic vector bundles over

E

(or even integrally), such as in work of Harris and Milne.

On the other hand, the parabolic subgroup

P_{μ}

is the stabilizer of the ascending filtration Fil. This gives rise to an antiholomorphic embedding

X ↪ Fl (C) = G (C) / P_{μ}

Morally,

P_{μ}

is the stabilizer of the Hodge-Tate filtration on the

p

-adic Ã©tale cohomology of

A_{K}

. The Hodge-Tate period morphism will be a

p

-adic analogue of the embedding (3.1) (or perhaps of the embedding (3.2), depending on one's perspective).

Let

p

be a rational prime,

p ∣ p

a prime of

E

, and let

C

be the completion of an algebraic closure of

E_{p}

. We consider the adic spaces

∮_{K}

and

F ℓ

over

Spa (C, O_{C})

corresponding to the algebraic varieties

S_{K}

and

F l

over

E

. A striking result of Scholze shows that the tower of Shimura varieties

{(S_{K^{p} K_{p}})}_{K_{p}}

acquires the structure of a perfectoid space (in the sense of [51]) as

K_{p}

varies over compact open subgroups of

G (Q_{p})

. More precisely, the following result was established in

[53, $ 3, 4]

and later refined in

[17, $ 2]

, by correctly identifying the target of the Hodge-Tate period morphism.

Theorem 3.1. There exists a unique perfectoid space

S_{K^{p}}

satisfying

S_{K^{p}} \sim \underset{lim_{K_{p}}}{} S_{K^{p} K_{p}},^{2}

in the sense of [55, DEFINITION 2.4.1], and a

G (Q_{p})

-equivariant morphism of adic spaces

π_{H T} : S_{K^{p}} \to F ℓ

Moreover,

π_{H T}

is equivariant for the usual action of Hecke operators away from

p

S_{K^{p}}

and their trivial action on

F ℓ

In the Siegel case

G = {G S p}_{2 n} / Q

, one can describe the Hodge-Tate period morphism

π_{H T}

from a moduli-theoretic perspective as follows. An abelian variety

A / C

, equipped with a trivialization

T_{p} A ≃ Z_{p}^{2 n}

will be sent to the first piece of the Hodge-Tate filtration

Lie

A \subset T_{p} A \otimes_{Z_{p}} C ≃ C^{2 n}

2 It is enough to consider the Shimura varieties as adic spaces over

E_{p}

and the tower still acquires a perfectoid structure in a noncanonical way. We work over

C

for simplicity and also because this gives rise to the Ã©tale cohomology groups we want to understand.

Dually, one has the Hodge-Tate filtration on the

p

-adic Ã©tale cohomology of

A

\begin{matrix} (3.3) & 0 \to H^{1} (A, O_{A}) \to H_{e t}^{1} (A, Z_{p}) \otimes_{Z_{p}} C \to H^{0} (A, Ω_{A / C}^{1}) (- 1) \to 0 \end{matrix}

where

(- 1)

denotes a Tate twist (which is important for keeping track of the Galois action). To show that the morphism defined this way on

Spa (C, C^{+})

-points comes from a morphism of adic spaces, it is important to know that the filtration (3.3) varies continuously. At the same time, to extend the result to Shimura varieties of Hodge type and to cut down the image to

F ℓ

, one needs to keep track of Hodge tensors carefully. Both problems are solved via relative

p

-adic Hodge theory for the morphism

A_{K} \to S_{K}

, where

A_{K}

is the restriction to

ς_{K}

of a universal abelian scheme over an ambient Siegel modular variety. See [13, Â§3] for an overview.

Theorem 3.1 can be extended to minimal and toroidal compactifications of Siegel modular varieties, cf. [53] and [49]. Moreover, there is a natural affinoid cover of

F ℓ

such that the preimage under

π_{H T}

of each affinoid in the cover is an affinoid perfectoid subspace of

S_{K^{p}}^{*}

. The underlying reason for this is the fact that the partial minimal compactification of the ordinary locus is affine. The perfectoid structure on

S_{K^{p}}^{*}

and the affinoid nature of the Hodge-Tate period morphism play an important role in Scholze's

p

-adic interpolation argument, that is key for the construction of Galois representations associated with torsion classes. See also [44] for an exposition of the main ideas.

Theorem 3.1 can also be extended to minimal and toroidal compactifications of Shimura varieties of Hodge type and even abelian type, cf. [32, 58] and [8], although there are some technical issues at the boundary. For example, the cleanest formulation currently available in full generality is that the relationship

S_{K^{p}}^{*} = lim_{⟵ K_{p}} S_{K^{p} K_{p}}^{*}

, for a perfectoid space

S_{K^{p}}^{*}

, holds in Scholze's category of diamonds [54].

Example 3.2. To see where the perfectoid structure on

S_{K^{p}}

comes from, it is instructive to consider the case of modular curves and study the geometry of their special fibers: we are particularly interested in the geometry of the so-called Deligne-Rapoport model. Set

G =

{G L}_{2} / Q

. Let

K_{p}^{0} = {G L}_{2} (Z_{p})

, the hyperspecial compact open subgroup and let

{\bar{S}}_{K^{p} K_{p}^{0}} / F_{p}

be the special fiber of the integral model over

Z_{(p)}

of the modular curve at this level. This is a smooth curve over

F_{p}

that represents a moduli problem

(E, α)

of elliptic curves equipped with prime-to-

p

level structures (determined by the prime-to-

p

level

K^{p}

). The isogeny class of the

p

-divisible group

E [p^{\infty}]

induces the Newton stratification

\begin{matrix} (3.4) & {\bar{S}}_{K^{p} K_{p}^{0}} = {\bar{S}}_{K^{p} K_{p}^{0}}^{o r d} ⊔ {\bar{S}}_{K^{p} K_{p}^{0}}^{s s} \end{matrix}

into an open dense ordinary stratum

{\bar{S}}_{K^{p} K_{p}^{0}}^{ord}

(where

E [p^{\infty}]

is isogenous to

μ_{p^{\infty}} \times Q_{p} / Z_{p}

) and a closed supersingular stratum

{\bar{S}}_{K^{p}}^{s s} K_{p}^{0}

consisting of finitely many points (where

E [p^{\infty}]

is connected).

Now let

K_{p}^{1} \subset {G L}_{2} (Q_{p})

be the Iwahori subgroup and

{\bar{S}}_{K^{p} K_{p}^{1}} / F_{p}

be the special fiber of the integral model of the modular curve at this level. This represents a moduli problem

(E, α, D)

of elliptic curves equipped with prime-to-

p

level structures and also with a level structure at

p

given by a finite flat subgroup scheme

D \subset E [p]

of order

p

. Again, we have the
preimage of the Newton stratification

{\bar{S}}_{K^{p} K_{p}^{1}} = {\bar{S}}_{K^{p} K_{p}^{1}}^{ord} ⊔ {\bar{S}}_{K^{p} K_{p}^{1}}^{s s}

. The modular curve at this level is not smooth, but rather a union of irreducible components that intersect transversely at the finitely many supersingular points.

The open and dense ordinary locus

{\bar{S}}_{K^{p} K_{p}^{1}}^{ord}

is a disjoint union of two KottwitzRapoport strata: the one where

D ≃ μ_{p}

and the one where

D ≃ Z / p Z

. Both of these Kottwitz-Rapoport strata can be shown to be abstractly isomorphic to the ordinary stratum at hyperspecial level. If we restrict the natural forgetful map

{\bar{S}}_{K^{p} K_{p}^{1}}^{ord} \to {\bar{S}}_{K^{p} K_{p}^{0}}^{ord}

to the Kottwitz-Rapoport stratum where

D ≃ Z / p Z

, the map can be identified (up to an isomorphism) with the geometric Frobenius. (The restriction of the map to the Kottwitz-Rapoport stratum where

D ≃ μ_{p}

is an isomorphism.)

On the adic generic fiber, one can extend this picture to an anticanonical ordinary tower, where the transition morphisms reduce modulo

p

to (powers of) the geometric Frobenius, giving a perfectoid space in the limit. To extend beyond the ordinary locus, Scholze uses the theory of the canonical subgroup, the action of

{G L}_{2} (Q_{p})

at infinite level, and a rudimentary form of the Hodge-Tate period morphism that is just defined on the underlying topological spaces.

The above strategy generalizes relatively cleanly to higher-dimensional Siegel modular varieties, modulo subtleties at the boundary. To extend Theorem 3.1 to general Shimura varieties of Hodge type, Scholze considers an embedding at infinite level into a Siegel modular variety. It is surprisingly subtle to understand directly the perfectoid structure on a general Shimura variety of Hodge type (especially in the case when

G_{Q_{p}}

is nonsplit) and this is related to the discussion in Section 5. This is also related to the fact that the geometry of the EKOR stratification is more intricate when

G_{Q_{p}}

is nonsplit.

For simplicity, let us now assume that

(G, X)

is a Shimura datum of PEL type and that

p

is an unramified prime for this Shimura datum. Recall the Kottwitz set

B (G)

classifying isocrystals with

G_{Q_{p}}

-structure. The Hodge cocharacter

μ

defines a subset

B (G, μ^{- 1}) \subset

B (G)

μ^{- 1}

-admissible elements. The special fiber of the Shimura variety with hyperspecial level at

p

admits a Newton stratification

{\bar{S}}_{K^{p} K_{p}^{0}} = ⨆_{b \in B (G, μ^{- 1})} {\bar{S}}_{K^{p} K_{p}^{0}}^{b}

into locally closed strata indexed by this subset. This stratification is in terms of isogeny classes of

p

-divisible groups with

G_{Q_{p}}

-structure and generalizes the stratification (3.4) from the modular curve case.

For each

b \in B (G, μ^{- 1})

, one can choose a (completely slope divisible)

p

-divisible group with

G_{Q_{p}}

-structure

X_{b} / {\bar{F}}_{p}

and define the corresponding Oort central leaf. This is a
smooth closed subscheme

C^{X b}

of the Newton stratum

{\bar{S}}_{K^{p} K_{p}^{0}}^{b}

, such that the isomorphism class of the

p

-divisible group with

G_{Q_{p}}

-structure over each geometric point of the leaf is constant and equal to that of

X_{b}

C^{X_{b}} = {x \in {\bar{S}}_{K^{p} K_{p}^{0}}^{b} ∣ {\bar{A}}_{K^{p} K_{p}^{0}} [p^{\infty}] \times κ (\bar{x}) ≃ X_{b} \times κ (\bar{x})}

In general, there can be infinitely many leaves inside a given Newton stratum. Over each central leaf, one has the perfect Igusa variety

{Ig}^{b} / {\bar{F}}_{p}

, a profinite cover of

C^{X^{b}}

which parametrizes trivializations of the universal

p

-divisible group with

G_{Q_{p}}

-structure.

Variants of Igusa varieties were introduced in [34] in the special case of Shimura varieties of Harris-Taylor type. They were defined more generally for Shimura varieties of PEL type by Mantovan [43] and their

ℓ

-adic cohomology was computed in many cases by Shin using a counting point formula [59-61]. All these authors consider Igusa varieties as profinite Ã©tale covers of central leaves, which trivialize the graded pieces of the slope filtration on the universal

p

-divisible group. Taking perfection gives a more elegant moduli-theoretic interpretation, while preserving

ℓ

-adic cohomology. However, the coherent cohomology of Igusa varieties is also important for defining and studying

p

-adic families of automorphic forms on

G

, as pioneered by Katz and Hida. Taking perfection is too crude for this purpose.

While the central leaf

C^{X_{b}}

depends on the choice of

X_{b}

in its isogeny class, one can show that the perfect Igusa variety

{I g}^{b}

only depends on the isogeny class: this follows from the equivalent moduli-theoretic description in [17, LEMMA 4.3.4] (see also [19, LEMMA 4.2.2], which keeps track of the extra structures more carefully). In particular, the pair

(G, μ)

is not determined by the Igusa variety

{I g}^{b}

- it can happen that Igusa varieties that are a priori obtained from different Shimura varieties are isomorphic. See [19, THEOREM 4.2.4] for an example and [57] for a systematic analysis of this phenomenon in the function field setting.

Because

{I g}^{b} / {\bar{F}}_{p}

is perfect, the base change

{I g}^{b} \times {\bar{F}}_{p} O_{C} / p

admits a canonical lift to a flat formal scheme over

Spf O_{C}

. We let

ℑ g^{b}

denote the adic generic fiber of this lift, which is a perfectoid space over

Spa (C, O_{C})

. The spaces

{I g}^{b}

and

ℑ g^{b}

have naturally isomorphic

ℓ

-adic cohomology groups and they both have an action of a locally profinite group

G_{b} (Q_{p})

, where

G_{b}

is an inner form of a Levi subgroup of

G

For each

b \in B (G, μ^{- 1})

, one can also consider the associated Rapoport-Zink space, a moduli space of

p

-divisible groups with

G_{Q_{p}}

-structure that is a local analogue of a Shimura variety. Concretely in the PEL case, one considers a moduli problem of

p

-divisible groups equipped with

G_{Q_{p}}

-structure, satisfying the Kottwitz determinant condition with respect to

μ

, and with a modulo

p

quasiisogeny to the fixed

p

-divisible group

X_{b}

. This moduli problem was shown by Rapoport-Zink [50] to be representable by a formal scheme over

Spf O_{{\overset{˘}{E}}_{p}}

, where

{\overset{˘}{E}}_{p}

is the completion of the maximal unramified extension of

E_{p}

. We let

M^{b}

denote the adic generic fiber of this formal scheme,

^{3}

base changed to

Spa (C, O_{C})

, and let

M_{\infty}^{b}

denote the corresponding infinite-level Rapoport-Zink space. The latter object can be shown to be a perfectoid space using the techniques of [55], by which the infinite-level Rapoport-Zink space admits a local analogue of the Hodge-Tate period morphism

π_{H T}^{b} : M_{\infty}^{b} \to F ℓ

It turns out that the geometry of

π_{H T}

is intricately tied up with the geometry of its local analogues

π_{H T}^{b}

. The following result is a conceptually cleaner, infinite-level version of the Mantovan product formula established in [43], which describes Newton strata inside Shimura varieties in terms of a product of Igusa varieties and Rapoport-Zink spaces.

Theorem 3.3. There exists a Newton stratification

F ℓ = ⨆_{b \in B (G, μ^{- 1})} F ℓ^{b}

into locally closed strata.

For each

b \in B (G, μ^{- 1})

, one can consider the Newton stratum

∮_{K^{p}}^{\circ b}

as a locally closed subspace of the good reduction locus

S_{K^{p}}^{\circ}

. There exists a Cartesian diagram of diamonds over

Spd (C, O_{C})

Moreover, each vertical map is a pro-Ã©tale torsor for the group diamond

{\tilde{G}}_{b}

(identified with

{Aut}_{G} ({\tilde{X}}_{b})

, in the notation of

[17, $ 4])

The decomposition into Newton strata is defined in [17, Â§3]. Morally, one first constructs a map of v-stacks

F ℓ \to

Bun

_{G}

, where the latter is the

v

-stack of

G

-bundles on the Fargues-Fontaine curve. To construct this map of v-stacks, it is convenient to notice that one can identify the diamond associated to

F ℓ

with the minuscule Schubert cell defined by

μ

inside the

B_{d R}^{+}

-Grassmannian for

G

. Once the map to

{Bun}_{G}

is in the picture, one uses Fargues's result that the points of Bun

_{G}

are in bijection with the Kottwitz set

B (G)

, cf. [27] (see also [2] for an alternative proof that also works in equal characteristic). Moreover, the Newton decomposition is a stratification, in the sense that, for

b \in B (G, μ)

, we have

\bar{F ℓ^{b}} = ⨆_{b^{'} \geq b} F ℓ^{b^{'}}

where

\geq

denotes the Bruhat order. The latter fact follows from a recent result of Viehmann, see [63, THEOREM 1.1].

On rank one points,

π_{H T}

is compatible with the two different ways of defining the Newton stratification: via pullback from

{\bar{S}}_{K^{p} K_{p}^{0}}

§_{K^{p}}

and via pullback from Bun

_{G}

F ℓ

. The behavior is more subtle on higher rank points. This is related to the fact that the closure relations are reversed in the two settings: the basic locus inside

{\bar{S}}_{K^{p} K_{p}^{0}}

is the unique closed stratum, whereas each basic stratum inside

{B u n}_{G}

is open. On the other hand, the

(μ)

-ordinary locus is open and dense inside

{\bar{S}}_{K^{p} K_{p}^{0}}

, whereas it is a zero-dimensional closed stratum inside

F ℓ

. The infinite-level product formula is established in [17, $4], although it is formulated in terms of functors on

{Perf}_{{\overset{˘}{E}}_{p}} \cdot^{4}

This was extended to Shimura varieties of Hodge type by Hamacher [31].

Assume that the Shimura varieties

S_{K}

are compact. We have the following consequence for the fibers of

π_{H T}

: let

\bar{x} : S p a (C, C^{+}) \to F ℓ^{b}

be a geometric point. Then there is an inclusion of

{ℑ g}^{b}

into

π_{H T}^{- 1} (\bar{x})

, which identifies the target with the canonical compactification of the source, in the sense of [54, PROPOSITION 18.6]. In [18, THEOREM 1.10], we extend the computation of the fibers to minimal and toroidal compactifications of (noncompact) Shimura varieties attached to quasisplit unitary groups. In this case, the fibers can be obtained from partial minimal and toroidal compactifications of Igusa varieties. It would be interesting to extend the whole infinite-level product formula to compactifications.

Example 3.4. We make the geometry of

π_{H T}

explicit in the case of the modular curve, i.e., for

G = {G L}_{2} / Q

. In this case, we identify

F ℓ = P^{1, ad}

and we have the decomposition into Newton strata

The ordinary locus inside

P^{1, ad}

consists of the set of points defined over

Q_{p}

and the basic / supersingular locus is its complement

Ω

, the Drinfeld upper half-plane.

The fibers of

π_{H T}

over the ordinary locus are "perfectoid versions" of Igusa curves. The infinite-level version of the product formula reduces, in this case, to the statement that the ordinary locus is parabolically induced from

ℑ g^{ord}

, as in

[19, $ 6]

. The fibers of

π_{H T}

over the supersingular locus are profinite sets: the corresponding Igusa varieties can be identified with double cosets

D^{\times} ∖ D^{\times} (A_{f}^{p}) / K^{p}

, where

D / Q

is the quaternion algebra ramified precisely at

\infty

and

p

. This precise result is established in [35], although the idea goes back to DeuringSerre. One should be able to give an analogous description for basic Igusa varieties in much greater generality - this is closely related to Rapoport-Zink uniformization.

4. COHOMOLOGY WITH MOD $ℓ$ COEFFICIENTS

In this section, we outline some recent strategies for computing the cohomology of Shimura varieties with modulo

ℓ

coefficients using the

p

-adic Hodge-Tate period morphism, where

ℓ

and

p

are two distinct primes. We emphasize the strategies developed in [17-19,38].

We will assume throughout that

(G, X)

is a Shimura datum of abelian type and, in practice, we will focus on two examples: the case of Shimura varieties associated with unitary similitude groups and the case of Hilbert modular varieties. Let

m \subset T

be a max- imal ideal in the support of

H_{(c)}^{*} (S_{K} (C), F_{ℓ})

. By work of Scholze (cf. [53, THEOREM 4.3.1]) and by the construction of Galois representations in the essentially self-dual case, we know in many cases how to associate a global modulo

ℓ

Galois representation

{\bar{ρ}}_{m}

to the maximal ideal

m

. Therefore, the non-Eisenstein condition makes sense, and one can at least formulate Conjecture 2.2. In order to make progress on this conjecture, we impose a local representation-theoretic condition at the prime

p

, which we treat as an auxiliary prime.

Definition 4.1. Let

F

be a finite field of characteristic

ℓ

(1) Let

p \neq ℓ

be a prime,

K / Q_{p}

be a finite extension, and

\bar{ρ} : Gal (\bar{K} / K) \to {G L}_{n} (F)

be a continuous representation. We say that

\bar{ρ}

is generic if it is unramified and the eigenvalues (with multiplicity)

α_{1}, \dots, α_{n} \in {\bar{F}}_{ℓ}

\bar{ρ} ({Frob}_{K})

satisfy

α_{i} / α_{j} \neq

| O_{K} / m_{K} |

for

i \neq j

(2) Let

F

be a number field and

\bar{ρ} : Gal (\bar{F} / F) \to {G L}_{n} (F)

be a continuous representation. We say that a prime

p \neq ℓ

is decomposed generic for

\bar{ρ}

p

splits completely in

F

and, for every prime

p ∣ p

F, {\bar{ρ} |}_{Gal ({\bar{F}}_{p} / F_{p})}

is generic. We say that

\bar{ρ}

is decomposed generic if there exists a prime

p \neq ℓ

which is decomposed generic for

\bar{ρ}

. (If one such prime exists, then infinitely many do.)

Remark 4.2. The condition for the local representation

\bar{ρ}

G a l (\bar{K} / K)

to be generic implies that any lift to characteristic 0 of

\bar{ρ}

corresponds under the local Langlands correspondence to a generic principal series representation of

{G L}_{n} (K)

. Such a representation can never arise from a nonsplit inner form of

{G L}_{n} / K

via the Jacquet-Langlands correspondence. For this reason, a generic

\bar{ρ}

cannot be the modulo

ℓ

reduction of the

L

-parameter of a smooth representation of a nonsplit inner form of

{G L}_{n} / K

A semisimple 2-dimensional representation

\bar{ρ}

Gal (\bar{Q} / Q)

is either decomposed generic or it satisfies (2.1): the case where

\bar{ρ}

is a direct sum of two characters can be analyzed by hand, and the case where

\bar{ρ}

is absolutely irreducible follows from the paragraph after Theorem 3.1 in [37]. More generally, the condition for a global representation

\bar{ρ}

Gal (\bar{F} / F)

to be decomposed generic can be ensured when

\bar{ρ}

has large image. For example, if

ℓ > 2, F

is a totally real field, and

\bar{ρ}

is a totally odd 2-dimensional representation with nonsolvable image, then

\bar{ρ}

is decomposed generic (cf. [19, LEMMA 7.1.8]).

Let

F

be an imaginary CM field. Let

(B, *, V, ⟨ \cdot, \cdot ⟩)

be a PEL datum of type

A

, where

B

is a central simple algebra with center

F

. We let

(G, X)

be the associated Shimura datum. For a neat compact open subgroup

K \subset G (A_{f})

, we let

S_{K} / E

be the associated Shimura variety, of dimension

d

. The following conjecture is a slightly different formulation of [38, CONJECTURE 1.2], with essentially the same content.

Conjecture 4.3. Let

n \subset T

be a maximal ideal in the support of

H_{(c)}^{i} (S_{K} (C), F_{ℓ})

. Assume that

{\bar{ρ}}_{m}

is decomposed generic. Then the following statements hold true:

(1) if

H_{c}^{i} {(S_{K} (C), F_{ℓ})}_{m} \neq 0

, then

i \leq d

;

(2) if

H^{i} {(S_{K} (C), F_{ℓ})}_{m} \neq 0

, then

i \geq d

If the Shimura varieties

S_{K}

are compact, or if we additionally assume

m

to be nonEisenstein, Conjecture 4.3 implies a significant part of Conjecture 2.2 for Shimura varieties of PEL type A. Analogues of Conjecture 4.3 can be formulated (and are perhaps within reach) for other Shimura varieties, such as Siegel modular varieties.

Theorem 4.4 ([17] strengthened in [38]). Assume that

G

is anisotropic modulo center, so that the Shimura varieties

S_{K}

are compact. Then Conjecture 4.3 holds true.

Theorem 4.5 ([18] strengthened in [38]). Assume that

B = F, V = F^{2 n}

and

G

is a quasisplit group of unitary similitudes. Then Conjecture 4.3 holds true.

Remark 4.6. The more recent results of [38] have significantly fewer technical assumptions than the earlier ones of [17] and [18]. For example, Koshikawa's version of Theorem 4.5 allows

F

to be an imaginary quadratic field. It seems nontrivial to obtain this case with the methods of [18]. In the noncompact case, his results rely on the geometric constructions in [18], in particular on the semiperversity result for Shimura varieties attached to quasisplit unitary groups that is established there. As he notes, a generalization of this semiperversity result should lead to a full proof of Conjecture 4.3 for Shimura varieties of PEL type A. The more general semiperversity result will be obtained in the upcoming PhD thesis of Mafalda Santos.

In the case of Harris-Taylor Shimura varieties, Theorem 4.4 was first proved by Boyer [9]. Boyer's argument uses the integral models of Shimura varieties of Harris-Taylor type, but it is close in spirit to the argument carried out in [17] on the generic fiber. What is really interesting about Boyer's results is that he goes beyond genericity, in the following sense. Given the eigenvalues (with multiplicity)

α_{1}, \dots, α_{n}

{\bar{ρ}}_{m} (

Frob

_{p})

, with

p ∣ p

the relevant prime of

F,^{5}

one can define a "defect" that measures how far

{\bar{ρ}}_{m}

is from being generic at

p

. Concretely, set

δ_{p} (m)

to be equal to the length of the maximal chain of eigenvalues where the successive terms have ratio equal to

| O_{F_{p}} / m_{F_{p}} |

. Boyer shows that the cohomology groups

H_{(c)}^{i} {(S_{K} (C), F_{ℓ})}_{m}

are nonzero at most in the range

[d - δ_{p} (m), d + δ_{p} (m)]

. As noted by both Emerton and Koshikawa, such a result is consistent with Arthur's conjectures on the cohomology of Shimura varieties with

C

-coefficients and points towards a modulo

ℓ

analogue of these conjectures.

Let us also discuss the analogous vanishing result in the Hilbert case. Let

F

be a totally real field of degree

g

and let

G = {Res}_{F / Q} {G L}_{2}

. For a neat compact open subgroup

K \subset G (A_{f})

, we let

S_{K} / Q

be the corresponding Hilbert modular variety, of dimension

g

Theorem 4.7 ([19, THEOREM A]). Let

ℓ > 2

and

m \subset T

be a maximal ideal in the support of

H_{(c)}^{i} (S_{K} (C), F_{ℓ})

. Assume that the image of

{\bar{ρ}}_{m}

is not solvable, which implies that

{\bar{ρ}}_{m}

is absolutely irreducible and decomposed generic. Then

H_{c}^{i} {(S_{K} (C), F_{ℓ})}_{m} = H^{i} {(S_{K} (C), F_{ℓ})}_{m}

is nonzero only for

i = g

5 In this special case, one does not have to impose the condition that

p

splits completely in

F

, and it suffices to have genericity at one prime

p ∣ p

The same result holds for all quaternionic Shimura varieties, and we can even prove the analogue of Boyer's result that goes beyond genericity in all these settings. As an application, we deduce (under some technical assumptions) that the

p

-adic local Langlands correspondence for

{G L}_{2} (Q_{p})

occurs in the completed cohomology of Hilbert modular varieties, when

p

is a prime that splits completely in

F

. This uses the axiomatic approach via patching introduced in [14] and further developed in [15,30].

We now outline the original strategy for proving Theorem 4.4, which was introduced in [17]. Let

p

be a prime and

K = K^{p} K_{p} \subset G (A_{f})

be a neat compact open subgroup. The Hodge-Tate period morphism gives rise to a

T

-equivariant diagram

The standard comparison theorems between various cohomology theories allow us to identify

H_{(c)}^{*} {(S_{K} (C), F_{ℓ})}_{m}

with

H_{(c)}^{*} {(S_{K}, F_{ℓ})}_{m}

. The arrow on the left-hand side of (4.1) is a

K_{p}

-torsor, so the Hochschild-Serre spectral sequence allows us to recover

H_{(c)}^{*} {(S_{K}, F_{ℓ})}_{m}

from

H_{(c)}^{*} {(ς_{K^{p}}, F_{ℓ})}_{m}

. The idea is now to compute

H_{(c)}^{*} {(ς_{K^{p}}, F_{ℓ})}_{m}

in two stages: first understand the complex of sheaves

{(R π_{H T *} F_{ℓ})}_{m}

F ℓ

, then compute the total cohomology using the Leray-Serre spectral sequence.

Two miraculous things happen that greatly simplify the structure of

{(R π_{H T *} F_{ℓ})}_{m}

. The first is that

{(R π_{H T *} F_{ℓ})}_{m}

behaves like a perverse sheaf on

F ℓ

. This is because

π_{H T}

is simultaneously affinoid, as discussed after Theorem 3.1, and partially proper, because the Shimura varieties were assumed to be compact. In particular, the restriction of

{(R π_{H T *} F_{ℓ})}_{m}

to a highest-dimensional stratum in its support is concentrated in one degree. By the computation of the fibers of

π_{H T}

, this implies that the cohomology groups

R Γ {(ℑ g^{b}, Z_{ℓ})}_{m}

are concentrated in one degree and torsion-free. The second miracle is that, whenever the group

G_{b} (Q_{p})

acting on

{s g}^{b}

comes from a nonquasisplit inner form, the localization

R Γ {({s g}^{b}, Q_{ℓ})}_{m}

vanishes. This uses the genericity of

{\bar{ρ}}_{m}

at each

p ∣ p

and suggests that the cohomology of Igusa varieties satisfies some form of local-global compatibility. Finally, the condition that

p

splits completely in

F

guarantees that the only Newton stratum for which

G_{b}

is quasisplit is the ordinary one. Therefore, the hypotheses of Theorem 4.4 guarantee that

{(R π_{H T *} F_{ℓ})}_{m}

is as simple as possible - it is supported in one degree on a zero-dimensional stratum!

The computation of

R Γ {({I g}^{b}, Q_{ℓ})}_{m}

, at least at the level of the Grothendieck group, can be done using the trace formula method pioneered by Shin [60]. This is the method used for Shimura varieties of PEL type A in [17] and [18]. For inner forms of

{Res}_{F / Q} {G L}_{2}

, with

F

a totally real field, one can avoid these difficult computations, cf. [19]. In this setting, one can reinterpret results of Tian-Xiao [62] on geometric instances of the Jacquet-Langlands correspondence as giving rise to exotic isomorphisms between Igusa varieties arising from different Shimura varieties. This is what happens for the basic stratum in Example 3.4. Then one can conclude by applying the classical Jacquet-Langlands correspondence.

In [38], Koshikawa introduces a novel and complementary strategy for proving these kinds of vanishing theorems. He shows that, under the same genericity assumption in Definition 4.1, only the restriction of

{(R π_{H T *} F_{ℓ})}_{m}

to the ordinary locus contributes to the total cohomology of the Shimura variety. To achieve this, he proves the analogous generic vanishing theorem for the cohomology

R Γ_{c} {(M^{b}, Z_{ℓ})}_{m_{p}}

of the Rapoport-Zink space, where

m_{p}

is a maximal ideal of the local spherical Hecke algebra at

p

. This relies on the recent work of Fargues-Scholze on the geometrization of the local Langlands conjecture [28].

Koshikawa's strategy is more flexible, allowing him to handle with ease the case where

F

is an imaginary quadratic field. On the other hand, the original approach also gives information about the complexes of sheaves

{(R π_{H T *} F_{ℓ})}_{m}

, rather than just about the cohomology groups

H_{(c)}^{*} {(S_{K} (C), F_{ℓ})}_{m}

. These complexes should play an important role for questions of local-global compatibility in Fargues's geometrization conjecture, cf. [26, 87].

5. COHOMOLOGY WITH MOD $p$ AND $p$ -ADIC COEFFICIENTS

The most general method for constructing

p

-adic families of automorphic forms from the cohomology of locally symmetric spaces is via completed cohomology. First introduced by Emerton in [24], this has the following definition:

{\tilde{H}}^{*} (K^{p}, Z_{p}) = lim_{n} \underset{\underset{K_{p}}{(l i m}}{(lim^{*}} (X_{K^{p} K_{p}}, Z / p^{n}))

where

K^{p} \subset G (A_{f})

is a sufficiently small, fixed tame level, and

K_{p} \subset G (Q_{p})

runs over compact open subgroups. This space has an action of the spherical Hecke algebra

T

, built from Hecke operators away from

p

, as well as an action of the group

G (Q_{p})

. One can make the analogous definition for completed cohomology with compact support, and a variant gives completed homology and completed Borel-Moore homology. See [25] for an excellent survey that gives motivation, examples, and sketches the basic properties of these spaces.

Motivated by heuristics from the

p

-adic Langlands programme, Calegari and Emerton made several conjectures about the range of degrees in which one can have nonzero completed (co)homology and about the codimension of completed homology over the completed group rings

Z_{p} [[K_{p}]]

. See [11, coNJECTURE 1.5] for the original formulation and [32, CONJECTURE 1.3] for a slightly different formulation, which emphasizes the natural implications between the various conjectures. In particular, Calegari-Emerton conjectured that

{\tilde{H}}_{c}^{i} (K^{p}, Z_{p}) = {\tilde{H}}^{i} (K^{p}, Z_{p}) = 0 for i > q_{0}

For Shimura varieties of preabelian type, the Calegari-Emerton conjectures were proved by Hansen-Johansson in [32], building on work of Scholze who proved the vanishing of completed cohomology with compact support for Shimura varieties of Hodge type [53].

We sketch Scholze's argument, which illustrates the role of

p

-adic geometry in this result. It is enough to show that

{\tilde{H}}_{c}^{i} (K^{p}, F_{p}) = lim_{K_{p}} H_{c}^{i} (S_{K^{p} K_{p}} (C), F_{p})

vanishes for

i > d = \dim_{E} S_{K}

. Since

(G, X)

is a Shimura datum of Hodge type, we are in the setting of Theorem 3.1 - in fact, we know that the minimal compactification

S_{K^{p}}^{*}

is perfectoid. The primitive comparison theorem of [52] gives an almost isomorphism between

{\tilde{H}}_{c}^{i} (K^{p}, F_{p}) \otimes O_{C} / p

and

H_{e t}^{i} (S_{K^{p}}^{*}, d^{+} / p)

, where

^{+} \subseteq O^{+}

is the subsheaf of sections that vanish along the boundary. On an affinoid perfectoid space, Scholze proved the almost vanishing of the Ã©tale cohomology of

O^{+} / p

in degree

i > 0

. With some care at the boundary, one deduces that it is enough to prove that the analytic cohomology groups

H_{a n}^{i} (S_{K^{p}}^{*}, L^{+} / p)

are almost 0 in degree

i > d

. This final step follows from a theorem of Scheiderer on the cohomological dimension of spectral spaces.

In [20] and [16], we study Shimura varieties with unipotent level at

p

. More precisely, assume that

(G, X)

is a Shimura datum of Hodge type and that

G_{Q_{p}}

is split. Choose a split model of

G

and a Borel subgroup

B

over

Z_{p}

, and let

U \subset B

be the unipotent radical.

Theorem 5.1 ([16, THEOREM 1.1]). Let

H \subseteq U (Z_{p})

be a closed subgroup. We have

lim_{K_{p} \supseteq H} H_{c}^{i} (S_{K^{p} K_{p}} (C), F_{p}) = 0 for i > d

This result is stronger than the Calegari-Emerton conjecture for completed cohomology with compact support, since we can take

H = {1}

and recover Scholze's result discussed above. In addition to the argument sketched above, the key new idea needed for Theorem 5.1 is that the Bruhat decomposition on the Hodge-Tate period domain

F ℓ

remembers how far different subspaces of

S_{K^{p} U (Z_{p})}^{*}

are from being perfectoid.

Example 5.2. Assume that

G = {G L}_{2} / Q

, so that we are working in the modular curve case. The Bruhat decomposition is given by

P^{1, ad} = A^{1, ad} ⊔ {\infty}

, with the two Bruhat cells in natural bijection with the two components of the ordinary locus in (3.5). We have a morphism of sites

π_{H T / U (Z_{p})} : {(S_{K}^{*} U (Z_{p}))}_{ét} \to | P^{1, a d} | / U (Z_{p})

where we take the quotient

| P^{1, a d} | / U (Z_{p})

only as a topological space. The preimage of

| A^{1, ad} | / U (Z_{p})

S_{K^{p} U (Z_{p})}

is a perfectoid space, as proved by Ludwig in [42]. The preimage of

| \infty | / U (Z_{p})

has a

Z_{p}

-cover that is an affinoid perfectoid space. This allows us to bound the support of each

R^{i} π_{H T * / U (Z_{p})} (L^{+} / p)

, and we conclude by the Leray spectral sequence.

More generally, the Bruhat decomposition

G = ⨆_{w \in W^{P μ}} B w P_{μ}

gives a decomposition

F ℓ = ⨆_{w \in W^{P_{μ}}} F ℓ^{w}

into locally closed Schubert cells indexed by certain Weyl group elements. For each

F ℓ^{w} / U (Z_{p})

, we can quantify how far its preimage in

S_{K^{p} U (Z_{p})}^{*}

is from being a perfectoid space, which depends on the length of the Weyl group element

w

. The assumption that

G_{Q_{p}}

is split guarantees that all the Weyl group elements lie in the ordinary locus inside

F ℓ

, which greatly simplifies the analysis. However, the analogue of Theorem 5.1 may hold even without the assumption that

G_{Q_{p}}

is split, and even when the ordinary locus is empty. There is some evidence in this direction, e.g., by using embeddings into higher-dimensional Shimura varieties attached to split groups, or by using the results of [36] to handle the Harris-Taylor case, as in the upcoming PhD thesis of Louis Jaburi.

The Bruhat decomposition on

F ℓ

has more recently been used by Boxer and Pilloni to define a version of higher Coleman theory indexed by each

w \in W^{P_{μ}}

in [8]. The development of higher Coleman and higher Hida theories shows that the geometric theory of

p

-adic automorphic forms on Shimura varieties is much richer than previously expected. Furthermore, the Bruhat decomposition indicates the form a

p

-adic Eichler-Shimura isomorphism should take, relating completed cohomology to these more geometric theories. In joint work in progress with Mantovan and Newton, we use the geometry described in Example 5.2 to give a new proof of the ordinary Eichler-Shimura isomorphism due to Ohta [46, 47]. Our result decomposes the ordinary completed cohomology of the modular curve in terms of Hida theory and higher Hida theory, the latter recently developed by Boxer and Pilloni in [7].

Theorem 5.1 seems far away from Conjecture 2.2, because it concerns Shimura varieties with "infinite level" at

p

. However, one could ask whether a version of Theorem 5.1 holds already at level

B (Z_{p})

, at least after applying an ordinary idempotent, as in Hida theory. If that were the case, the control theorems in Hida theory (specifically the result known as independence of level) and a careful application of PoincarÃ© duality would imply that an

ℓ = p

analogue of Conjecture 4.3 holds, with generic replaced by ordinary. More precisely, in this case, the "auxiliary prime"

p

where we impose a representation-theoretic condition is no longer auxiliary but rather equal to

ℓ

6. APPLICATIONS BEYOND SHIMURA VARIETIES

While the focus of this article has been the cohomology of Shimura varieties, Theorems 4.5 and 5.1 have surprising applications to understanding the cohomology of more general locally symmetric spaces. For example, let

F

be an imaginary

C M

field and

G =

{Res}_{F / Q} {G L}_{n}

. Then

G

can be realized as the Levi quotient of the Siegel maximal parabolic of a quasisplit unitary group

\tilde{G}

. The Borel-Serre compactification

{\tilde{X}}_{\tilde{K}}^{B S}

for the locally symmetric spaces associated with the unitary group

\tilde{G}

gives rise to a Hecke-equivariant long exact sequence of the form

\begin{aligned} \dots & \to H_{c}^{i} ({\tilde{X}}_{\tilde{K}}, Z / ℓ^{n} Z) \to H^{i} ({\tilde{X}}_{\tilde{K}}, Z / ℓ^{n} Z) \to H^{i} (\partial {\tilde{X}}_{\tilde{K}}, Z / ℓ^{n} Z) \\ (6.1) & \to H_{c}^{i + 1} ({\tilde{X}}_{\tilde{K}}, Z / ℓ^{n} Z) \to \dots \end{aligned}

where

\partial {\tilde{X}}_{\tilde{K}} = {\tilde{X}}_{\tilde{K}}^{B S} ∖ {\tilde{X}}_{\tilde{K}}

is the boundary of the Borel-Serre compactification. The usual and compactly supported cohomology of

{\tilde{X}}_{\tilde{K}}

can be simplified to some extent by applying either of the two vanishing theorems. On the other hand, the cohomology of

X_{K}

can be shown to contribute to the cohomology of

\partial {\tilde{X}}_{\tilde{K}}

, in some more or less controlled fashion.

Let

m \subset T

be a non-Eisenstein maximal ideal in the support of

R Γ (X_{K}, Z_{ℓ})

and let

T (K)_{m}

denote the quotient of

T

that acts faithfully on

R Γ {(X_{K}, Z_{ℓ})}_{m}

. In addition to the residual Galois representation

{\bar{ρ}}_{m}

, Scholze associates to

m

a deformation

ρ_{m}

valued in

T (K)_{m} / I

, for some nilpotent ideal

I

. This was subsequently shown by Newton and Thorne in [45] to satisfy

I^{4} = 0

. In [20], we used a variant of Theorem 5.1 together with the excision sequence (6.1) to eliminate this nilpotent ideal entirely, under the assumption that

ℓ

splits
completely in the CM field

F

. This leads to a more natural statement on the existence of Galois representations in this setting.

The Galois representations

ρ_{m}

are expected to satisfy a certain property known as local-global compatibility, which is particularly subtle to state and prove at primes above

ℓ

. For example, after inverting

ℓ

, the

ρ_{m}

are expected to be de Rham, in the sense of Fontaine, but it is less clear what the right condition should be for torsion Galois representations. In another application, Theorem 4.5 is crucially used in [1] together with the excision sequence (6.1) to prove that

ρ_{m}

satisfies the expected local-global compatibility at primes above

ℓ

in two restricted families of cases: the ordinary case and the FontaineLaffaille case.

^{6}

In joint work in progress with Newton, we should be able to extend these methods to cover significantly more.

The local-global compatibility results established in [1] are already extremely useful: they help us implement the Calegari-Geraghty method unconditionally for the first time in arbitrary dimension. A striking application is the following result.

Theorem 6.1 ([1, THEOREM 1.0.1]). Let

F

be a CM field and

E / F

be an elliptic curve that does not have complex multiplication. Then

E

is potentially automorphic and satisfies the Sato-Tate conjecture.

The potential automorphy of

E

was established at the same time in work of BoxerCalegari-Gee-Pilloni [6], who also showed the potential automorphy of abelian surfaces over totally real fields. Their work relies on the Calegari-Geraghty method for the coherent cohomology of Shimura varieties and uses a preliminary version of higher Hida theory, due to Pilloni, as a key ingredient.

ACKNOWLEDGMENTS

I am grateful to my many colleagues and collaborators who have discussed mathematics in general and Shimura varieties in particular with me - I especially want to thank Frank Calegari, Matthew Emerton, Toby Gee, Dan Gulotta, Pol van Hoften, Christian Johansson, Elena Mantovan, Sophie Morel, James Newton, Vytas PaÅ¡kÅ«nas, Vincent Pilloni, Peter Scholze, Sug Woo Shin, Matteo Tamiozzo, and Richard Taylor. I am grateful to Toby Gee, Louis Jaburi, Teruhisa Koshikawa, Peter Scholze, and Matteo Tamiozzo for their comments on an earlier draft of this article.

FUNDING

This work was partially supported by a Royal Society University Research Fellowship, by a Leverhulme Prize, and by ERC Starting Grant No. 804176.

6 Up to possibly enlarging the nilpotent ideal

I

. It is not clear how to remove the nilpotent ideal from the statement of local-global compatibility at

ℓ = p

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ANA CARAIANI

180 Queen's Gate, London SW7 2AZ, United Kingdom, a.caraiani @ imperial.ac.uk

SAMIT DASGUPTA AND MAHESH KAKDE

Abstract

We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s = 1$ (equivalently, $s = 0$ ). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form.

The conjecture considers a totally real field $F$ and a finite abelian CM extension $H / F$ . It states the existence of $p$ -units in $H$ whose valuations at places above $p$ are related to the special values of the $L$ -functions of the extension $H / F$ at $s = 0$ . Essentially equivalently, the conjecture states that a Stickelberger element associated to $H / F$ annihilates the (appropriately smoothed) class group of $H$ .

We describe our recent proofs of the Brumer-Stark conjecture away from 2. The conjecture has been refined by many authors in multiple directions. We state some of these refinements and our results towards them. The key technique involved in the proofs is Ribet's method.

One of the refinements we discuss is an exact $p$ -adic analytic formula for Brumer-Stark units stated by the first author and his collaborators. We describe this formula and highlight some salient points of its proof. Since the Brumer-Stark units along with other easily described elements generate the maximal abelian extension of a totally real field, our results can be viewed as an explicit class theory for such fields. This can be considered a $p$ -adic version of Hilbert's 12th problem.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11R37; Secondary 11R42

KEYWORDS

Stark conjectures, Brumer-Stark units, Explicit class field theory

1. BACKGROUND AND MOTIVATION

Dirichlet's class number formula, conjectured for quadratic fields by Jacobi in 1832 and proven by Dirichlet in 1839, is one of the earliest examples of a relationship between leading terms of

L

-functions and global arithmetic invariants. Let

F

be a number field with ring of integers

O_{F}

. The Dedekind zeta function associated with

F

is defined as

ζ_{F} (s) = \sum_{0 \neq a \subset O_{F}} {N a}^{- s}, Re (s) > 1

where a runs through the nonzero ideals in

O_{F}

. The function

ζ_{F} (s)

generalizes Riemann's zeta function and has a meromorphic continuation to the whole complex plane with only a simple pole at

s = 1

. Dirichlet's class number formula, which is proved using a "geometry of numbers" approach, evaluates the residue at

s = 1

lim_{s \to 1} (s - 1) ζ_{F} (s) = \frac{2^{r_{1}} (2 π)^{r_{2}} R_{F} h_{F}}{w_{F} \sqrt{| D_{F} |}}

Here

r_{1}

is the number of real embeddings of

F

and

2 r_{2}

is the number of complex embeddings of

F

. Further,

h_{F}

and

R_{F}

denote the class number and regulator (defined below) of

F

, respectively, while

w_{F}

denotes the number of roots of unity in

F

and

D_{F}

is the discriminant of

F / Q

. The meromorphic function

ζ_{F} (s)

satisfies a functional equation relating

ζ_{F} (s)

and

ζ_{F} (1 - s)

. Using this functional equation, Dirichlet's class number formula can be restated as giving the leading term of the Taylor expansion of

ζ_{F} (s)

s = 0

\begin{matrix} (1.1) & ζ_{F} (s) = - \frac{h_{F} R_{F}}{w_{F}} s^{r_{1} + r_{2} - 1} + O (s^{r_{1} + r_{2}}) \end{matrix}

Artin described a theory of

L

-functions generalizing the Dedekind zeta function. Let

G_{F}

be the absolute Galois group of

F

. A Dirichlet character for

F

(or a degree 1 Artin character of

F

) is a homomorphism

χ : G_{F} \to C^{\times}

with finite image. Class field theory identifies

χ

with a function, again denoted by

χ

, from the set of nonzero ideals of

O_{F}

C

. Define

L (χ, s) = \sum_{0 \neq a \subset O_{F}} χ (a) {N a}^{- s}, Re (s) > 1

Again

L (χ, s)

has a meromorphic continuation to the whole complex plane with only a simple pole at

s = 1

χ

is trivial. If

H / F

is a Galois extension with finite abelian Galois group

G = Gal (H / F)

, then we can view any character

χ \in \hat{G} = Hom (G, C^{*})

as a Dirichlet character for

F

, and we have the Artin factorization formula

\begin{matrix} (1.2) & ζ_{H} (s) = \prod_{χ \in \hat{G}} L (χ, s) \end{matrix}

Dirichlet's class number formula (1.1) for the field

H

gives the leading term of the left-hand side of (1.2) at

s = 0

. Stark asked for an analogous formula for

L (χ, s)

s = 0

for each character

χ

, thereby giving a canonical factorization of the term

h_{H} R_{H} / w_{H}

. This led to the formulation of the abelian Stark conjecture, which we state in Section 2. This statement involves the choice of places of

F

that split completely in

H

. After stating Stark's conjecture, we restrict in the remainder of the paper to the case that the splitting places of

F

are finite.

Since the associated

L

-values here are algebraic, one can make progress on the conjectures through

p

-adic techniques such as

p

-adic Galois cohomology. To obtain nonzero

L

-values (and hence have nontrivial statements), parity conditions force us to restrict to the setting that

F

is a totally real field and

H

is a CM field.

Stark's conjecture at finite places has a natural restatement in terms of annihilators of class groups as formulated in the Brumer-Stark conjecture. We recall the statement and its refinements in Sections 4-5. The rest of the paper is taken up in describing the statement and proofs of our results toward the Brumer-Stark conjecture and its refinements.

2. STARK'S CONJECTURE

Let us first reformulate Dirichlet's class number formula.

For any place

w

F

we normalize the absolute value

| \cdot |_{w} : F_{w}^{*} \to R

| u |_{w} = {\begin{cases} | u | & if w is real \\ | u |^{2} & if w is complex \\ N w^{- {ord}_{w} (u)} & if w is nonarchimedean. \end{cases}

For a finite set of places

S

F

, let

X_{S}

denote the degree zero subgroup of the free abelian group on

S

. Let

u_{1}, \dots, u_{r_{1} + r_{2} - 1}

be a set of generators of the free abelian group

O_{F}^{*} / μ_{F}

. Let

S_{\infty}

be the set of archimedean places of

F

The Dirichlet regulator map

O_{F}^{*} / μ_{F} \to R X_{S_{\infty}}, u \mapsto \sum_{w \in S_{\infty}} \log | u |_{w} \cdot w

induces an isomorphism

R O_{F}^{*} \to R X_{S_{\infty}}

. Here and throughout,

R X_{S_{\infty}}

denotes

R \otimes_{Z} X_{S_{\infty}}

, etc. Let

w_{1}, \dots, w_{r_{1} + r_{2}}

denote the archimedean places of

F

. Then

\begin{matrix} (2.1) & {w_{i} - w_{1} : 2 \leq i \leq r + 1}, r = r_{1} + r_{2} - 1 \end{matrix}

is an integral basis of

X_{S_{\infty}}

. Let

R_{F}

be the absolute value of the determinant of the isomorphism between

R O_{F}^{*}

and

R X_{S_{\infty}}

with respect to the bases

{u_{1}, \dots, u_{r_{1} + r_{2} - 1}}

and (2.1), respectively. Up to a sign, Dirichlet's class number formula can be restated as follows:

(i) The rational structure

Q O_{F}^{*}

on the left-hand side corresponds to the structure

ζ_{F}^{(r)} (0) Q X_{S_{\infty}}

on the right-hand side.

(ii) The integral structure

O_{F}^{*} / μ_{F}

on the left-hand side corresponds to the structure

ζ_{F}^{(r)} (0) X_{S_{\infty}}

on the right-hand side.

Motivated by this reformulation, we present Stark's conjecture and its integral refinement due to Rubin. For details, see [41]. Let

F

be a number field of degree

n

and let

H / F

be a finite Galois extension with

G = Gal (H / F)

abelian. Let

S, T

be two finite disjoint sets of places of

F

satisfying the following conditions:

(1)

S

contains the sets

S_{\infty}

of archimedean places and

S_{ram}

of places ramified in

H

(2)

T

contains at least two primes of different residue characteristic or at least one prime of residue characteristic larger than

n + 1

, where

n = [F : Q]

For any character

χ \in \hat{G} = Hom (G, C^{*})

, define the

S

-depleted,

T

-smoothed

L

-function

L_{S, T} (χ, s) = \prod_{p \notin S} \frac{1}{1 - χ (p) {N p}^{- s}} \prod_{p \in T} (1 - χ (p) {N p}^{1 - s}), Re (s) > 1

The function

L_{S, T} (χ, s)

extends by analytic continuation to a holomorphic function on

C

. The Stickelberger element associated to this data is the unique group-ring element

Θ_{S, T} (H / F, s) \in C [G]

satisfying

χ (Θ_{S, T} (H / F, s)) = L_{S, T} (χ^{- 1}, s) for all χ \in \hat{G}

Let

S_{H}

denote the set of places of

H

above those in

S

, and similarly for

T_{H}

. Define

U_{S, T} = {x \in H^{*} : {ord}_{w} (x) \geq 0 for all w \notin S_{H} and x \equiv 1 (mod T_{H})}

The condition on

T

ensures that

U_{S, T}

does not have any torsion. The Galois equivariant version of Dirichlet's unit theorem gives an

R [G]

-module isomorphism

\begin{aligned} λ : R U_{S, T} \to R X_{S_{H}} \\ (2.2) & λ (u) = \sum_{w \in S_{H}} \log (| u |_{w}) \cdot w \end{aligned}

Suppose that exactly

r

places

v_{1}, \dots, v_{r} \in S

split completely in

H

and

# S \geq r + 1

. The order of vanishing of

L_{S, T} (χ, s)

s = 0

is given by

r (χ) = \dim_{C} {(C U_{S, T})}^{(χ)} = {\begin{cases} # {v \in S : χ (v) = 1} & if χ \neq 1 \\ # S - 1 & if χ = 1 \end{cases}

whence

r (χ) \geq r

for all

χ \in \hat{G}

. Stark's conjecture predicts that the

r

th derivative

Θ_{S, T}^{(r)} (H / F, 0)

captures the "non-rationality" of the map

λ

Conjecture 2.1 (Stark). We have

Θ_{S, T}^{(r)} (H / F, 0) \cdot Q \overset{r}{⋀} X_{S_{H}} \subset λ (Q \overset{r}{⋀} U_{S, T})

Concretely, this states that for each character

χ

G

with

r (χ) = r

, the nonzero number

L_{S, T}^{(r)} (χ^{- 1}, s)

lies in the one-dimensional

Q

-vector space spanned by

λ (\overset{r}{⋀} (U_{S, T}^{(χ)}))

Let us reformulate Conjecture 2.1 in terms of the existence of special elements. Write

X_{S_{H}}^{*} = Hom (X_{S_{H}}, Z [G])

. For

φ \in \land^{r} X_{S_{H}}^{*}

, there is a determinant map

\overset{r}{⋀} X_{S_{H}} \times \overset{r}{⋀} X_{S_{H}}^{*} \to Z [G]

defined by

(x_{1} \land \dots \land x_{r}, φ_{1} \land \dots \land φ_{r}) \mapsto φ_{1} \land \dots \land φ_{r} (x_{1} \land \dots \land x_{r}) = \det {(φ_{i} (x_{j}))}_{i, j}

We extend the determinant map to

R

-linearizations. We fix a place

w_{i}

H

above each

v_{i}

. Let

w_{i}^{*} \in X_{S_{H}}^{*}

be induced by

w_{i}^{*} (w) = \sum_{γ \in G : γ w_{i} = w} γ

Conjecture 2.2 (Stark). Put

φ = w_{1}^{*} \land \dots \land w_{r}^{*}

. There exists

u \in Q \land^{r} U_{S, T}

such that

φ (λ (u)) = Θ_{S, T}^{(r)} (H / F, 0)

The equivalence of Conjectures 2.1 and 2.2 is proven in [41, PROPOSITION 2.4].

We are now ready to state the integral version of Stark's conjecture. In the rank

r = 1

case, Stark proposed the statement that

u

in Conjecture 2.2 lies in

U_{S, T}

. This is the famous "rank 1 abelian Stark conjecture." In the higher rank case, the obvious generalization

u \in \overset{r}{⋀} U_{S, T}

is not true, as was experimentally observed by Rubin [41]. Rubin defined a lattice, nowadays called "Rubin's lattice" and conjectured that it contains the element

u

Put

U_{S, T}^{*} = {Hom}_{Z [G]} (U_{S, T}, Z [G])

The

r

th exterior bidual of

U_{S, T}

(see [7] for a more general study and the initiation of this terminology) is defined by

⋂^{r} U_{S, T} = {(\overset{r}{⋀} U_{S, T}^{*})}^{*} ≅ {x \in \overset{r}{⋀} Q U_{S, T} : φ (x) \in Z [G] for all φ \in \overset{r}{⋀} U_{S, T}^{*}}

We would like to consider only the "rank

r

" component of this bidual. To this end, for each character

χ \in \hat{G}

consider the associated idempotent

e_{χ} = \frac{1}{# G} \sum_{g \in G} χ (g) g^{- 1} \in C [G]

Define

e_{r} = \sum e_{χ} \in Q [G]

, where the sum extends over the set

{χ \in \hat{G} : L_{S, T}^{(r)} (χ, 0) \neq 0} = {χ \in \hat{G} : χ (G_{v}) \neq 1, v \in S ∖ {v_{1}, \dots, v_{r}}}

Define Rubin's lattice by

L^{(r)} U_{S, T} = (⋂^{r} U_{S, T}) \cap e_{r} (Q \overset{r}{⋀} U_{S, T}^{*})

The following is Rubin's higher rank integral Stark conjecture.

Conjecture 2.3 ([41], Conjecture B'). Put

φ = w_{1}^{*} \land \dots \land w_{r}^{*}

. There exists

u \in L^{(r)} U_{S, T}

such that

φ (λ (u)) = Θ_{S, T}^{(r)} (H / F, 0)

3. STARK'S CONJECTURES AT FINITE PLACES

We now assume that the totally split places

v_{1}, \dots, v_{r}

from the previous section are all finite. This happens only when

F

is a totally real field and

H

is totally complex. In fact, the fixed fields of characters with nonvanishing

L

-functions at 0 are

C M

fields, so we restrict to the setting where

F

is totally real and

H

C M

for the remainder of the article. We also
enact a slight notational change and write the set denoted

S

in the previous sections as

S^{'}

, and let

S = S^{'} ∖ {v_{1}, \dots, v_{r}}

. The reason for this is that we now still have

S \supset S_{\infty} \cup S_{ram}

As we explain, in this setting Conjecture 2.2 for

S^{'}

follows from a classical rationality result of Klingen-Siegel, though the integral refinement in Conjecture 2.3 remains a nontrivial statement. For a fixed place

w

H

, we have

\begin{matrix} (3.1) & \log | u |_{w} = - {ord}_{w} (u) \log N w \end{matrix}

Since the Euler factors at the

v_{i}

are equal to

(1 - N v_{i}^{- s}) = (1 - N w_{i}^{- s})

, we also have

\begin{matrix} (3.2) & Θ_{S^{'}, T}^{(r)} (H / F, 0) = Θ_{S, T} (H / F, 0) \cdot \prod_{i = 1}^{r} \log N w_{i} \end{matrix}

Theorem 3.1 (Klingen-Siegel). We have

Θ_{S, T} := Θ_{S, T} (H / F, 0) \in Q [G]

With

e_{r}

as in the previous section, we are then led to define a map over

Q

λ_{Q} : e_{r} (Q U_{S^{'}, T}) \to e_{r} (Q X_{S_{H}^{'}}), λ_{Q} (u) = \sum_{w ∣ v_{i} some i} {ord}_{w} (u) \cdot w

Note that

e_{r} (Q X_{S_{H}^{'}})

is the

Q

-vector space generated by the places of

H

above the

v_{i}

. The map

λ_{Q}

is a

Q [G]

-module isomorphism, and it induces an isomorphism on the free rank one

Q [G]

-modules obtained by taking

r

th wedge powers. In view of (3.1), the map on

r

th wedge powers induced by the map

λ

of (2.2), when restricted to

e_{r} (Q \land^{r} U_{S^{'}, T})

, is equal to

(\prod_{i = 1}^{r} \log N w_{i}) \cdot λ_{Q}

. Conjecture 2.2 follows from this observation together with (3.2), since Theorem 3.1 implies the existence of

u \in e_{r} (Q \overset{r}{⋀} U_{S^{'}, T})

such that

φ (λ_{Q} (u)) = Θ_{S, T}

Here

φ = w_{1}^{*} \land \dots \land w_{r}^{*}

as in the statement of the conjecture.

On the other hand, the integral statement in Conjecture 2.3 lies deeper. We first note the following celebrated theorem of Deligne-Ribet [21] and Cassou-NoguÃ¨s [8] refining the Klingen-Siegel theorem. The condition on the set

T

stated in Section 2 is crucial in this result. We remark that Deligne-RIbet prove their result using Hilbert modular forms, as an integral refinement of the strategy of the strategy established earlier by Siegel. This theme reappears in our own work described in

§ 6

Theorem 3.2. We have

Θ_{S, T} \in Z [G]

Conjecture 2.3 in this setting is known as the Rubin-Brumer-Stark conjecture:

Conjecture 3.3 (Rubin-Brumer-Stark). There exists

u \in L^{(r)} U_{S^{'}, T}

such that

φ (λ_{Q} (u)) = Θ_{S, T}

We describe in Theorem 4.3 below a strong partial result toward this conjecture.

4. THE BRUMER-STARK CONJECTURE

Having stated the higher rank Rubin-Brumer-Stark conjecture, we now wind back the clock and focus on the case

r = 1

. This case had been studied independently by Brumer and Stark and served as a motivation for Rubin's work. Writing the splitting prime

v_{1}

p

, the conjecture may be stated as follows.

Conjecture 4.1 (Brumer-Stark). Fix a prime ideal

p \subset O_{F}, p \notin S \cup T

, such that

p

splits completely in

H

. Fix a prime

P \subset O_{H}

above

p

. There exists a unique element

u_{p} \in H^{*}

such that

{| u_{p} |}_{w} = 1

for every place

w

H

not lying above

p

{ord}_{G} (u_{p}) := \sum_{σ \in G} {ord}_{β} (σ (u_{p})) σ^{- 1} = Θ_{S, T}

and

u \equiv 1 (mod q)

for all

q \in T_{H}

Note that the condition

| u |_{w} = 1

includes all complex places

w

, so

c (u_{p}) = u_{p}^{- 1}

for the unique complex conjugation

c \in G

As we have alread noted, Stark arrived upon this statement in the 1970s through his attempts to generalize and factorize the classical Dirichlet class number formula (though in a slightly different formulation; the statement above is due to Tate [46]). Prior to this, in the 1960s, Brumer was interested in generalizing Stickelberger's classical factorization formula for Gauss sums in cyclotomic fields. Stickelberger's result can be formulated as stating that when

H = Q (μ_{N})

is a cyclotomic field, the Stickelberger element annihilates the class group of

H

. Let us consider Brumer's perspective of annihilation of class groups in the case of general

H / F

4.1. Annihilation of class groups

Let

{C l}^{T} (H)

denote the ray class group of

H

with conductor equal to the product of primes in

T_{H}

. This is defined as follows. Let

I_{T} (H)

denote the group of fractional ideals of

H

relatively prime to the primes in

T_{H}

. Let

P_{T} (H)

denote the subgroup of

I_{T} (H)

generated by principal ideals

(α)

where

α \in O_{H}

satisfies

α \equiv 1 (mod q)

for all

q \in T_{H}

. Then

{C l}^{T} (H) = I_{T} (H) / P_{T} (H)

This

T

-smoothed class group is naturally a

G

-module.

With the notation as in Conjecture 4.1, we have

\begin{matrix} (4.2) & B^{Θ_{S, T}} = (u_{p}) \end{matrix}

Such an equation holds for all

p \notin S \cup T

that split completely in

H

. The set of primes of

H

above all such

p

generate

{C l}^{T} (H)

. Hence we deduce

\begin{matrix} (4.3) & Θ_{S, T} \in {Ann}_{Z [G]} ({C l}^{T} (H)) \end{matrix}

In fact, (4.3) is almost equivalent to Conjecture 4.1; given (4.2), the element

u_{p}

satisfies all of the conditions necessary for Conjecture 4.1 except possibly

c (u_{p}) = u_{p}^{- 1}

. But of course

v_{p} = u_{p} / c (u_{p})

satisfies this condition and moreover satisfies

P^{2 Θ_{S, T}} = (v_{p})

. Therefore the
only possible discrepancy between the statements is a factor of 2 , which disappears when we localize away from 2 as in the rest of this paper. Let us therefore define

R = Z [1 / 2] [G]^{-} = Z [1 / 2] [G] / (c + 1)

and for any

Z [G]

-module

M

we write

M^{-} = M \otimes_{Z [G]} R

. There exists an element

u_{p} \in O_{H} [1 / p]^{*} \otimes Z [1 / 2]

satisfying Conjecture 4.1 if and only if

\begin{matrix} (4.4) & Θ_{S, T} \in {Ann}_{R} ({C l}^{T} (H)^{-}) \end{matrix}

This is the Brumer-Stark conjecture "away from 2".

Many authors have studied (4.4) as well as refinements. The works of Burns, Greither, Kurihara, Popescu, and Sano are particularly noteworthy [4-7, 25-27,35]. Many of these refinements involve Fitting ideals, whose definition we now recall.

Let

R

be a commutative ring and

M

R

-module with finite presentation:

R^{m} \overset{A}{\to} R^{n} \to X \to 0

Here

A

is an

n \times m

matrix over

R

. The

i

th Fitting ideal Fitt

i, R (M)

is the ideal generated by the

n - i \times n - i

minors of

A

. It is a standard fact [37, CHAPTER 3, THEOREM 1] that Fitt

_{i, R} (M)

does not depend on the chosen presentation of

M

. We write

{Fitt}_{R} (M)

for

{Fitt}_{0, R} (M)

, and when these is no ambiguity about the choice of

i = 0

, we call this the Fitting ideal of

M

The Fitting ideal of a finitely presented module is contained in its annihilator:

\begin{matrix} (4.5) & {Fitt}_{R} (M) \subset {Ann}_{R} (M) \end{matrix}

In view of (4.4) and (4.5), it is therefore natural to ask whether

Θ_{S, T}

lies in the Fitting ideal of

{C l}^{T} (H)^{-}

over

R

. It was noticed by Popescu in the function field case [38] and by Kurihara in the number field case that while this holds sometimes, it does not always hold. Greither and Kurihara

[25, 26]

observed that the statement may be corrected by replacing

{C l}^{T} (H)^{-}

by its Pontryagin dual

{C l}^{T} (H)^{-, \lor} = {Hom}_{Z} ({C l}^{T} (H)^{-}, Q / Z)

We endow

{C l}^{T} (H)^{-, \lor}

with the contragradient

G

-action

g \cdot φ (x) = φ (g^{- 1} x)

. Denote by # the involution on

Z [G]

induced by

g \mapsto g^{- 1}

for

g \in G

Conjecture 4.2 (Kurihara, "Strong Brumer-Stark"). We have

Θ_{S, T}^{#} \in {Fitt}_{R} ({C l}^{T} (H)^{-, v})

Conjecture 4.2 leads to the following natural questions:

(1) What is the Fitting ideal of

{C l}^{T} (H)^{-, v}

(2) What is the Fitting ideal of

{C l}^{T} (H)^{-}

(3) Is there a natural arithmetically defined

R

-module whose Fitting ideal is generated by

Θ_{S, T}

Θ_{S, T}^{#}

The precise conjectural description of the Fitting ideal of

{C l}^{T} (H / F)^{-, \lor}

was given by Kurihara [35]; we state this in Section 5.1 below. An important fact about this statement is that when

S_{ram}

is nonempty, the Fitting ideal of

{C l}^{T} (H / F)^{-, \lor}

is in general not principal (and in particular is not generated by

Θ_{S, T}^{#}

A conjectural answer to the second question above has recently been provided in a striking paper by Atsuta and Kataoka [1]. They show that their conjecture is implied by the Equivariant Tamagawa Number Conjecture.

The third question is answered by a conjecture of Burns, Kurihara, and Sano, and is the topic of Section

§ 5.3

. We note that Fitting ideals of finitely presented

R

-modules are rarely principal. It is therefore remarkable that Burns-Kurihara-Sano defined a natural arithmetic object whose Fitting ideal is principal.

4.2. Our results

We now describe some of our results toward these conjectures [17, THEOREM 1.4].

Theorem 4.3. Kurihara's exact formula for

{F i t t}_{R} ({C l}^{T} (H)^{-, \lor})

holds (see Theorem 5.1). In particular, we have the Brumer-Stark and Strong Brumer-Stark conjectures away from 2:

Θ_{S, T}^{#} \in {Fitt}_{R} ({C l}^{T} (H)^{-, v}) \subset {Ann}_{R} {({C l}^{T} (H)^{-})}^{#}

Finally, Rubin's higher rank Brumer-Stark conjecture holds away from 2: with notation as in Conjecture 3.3, there exists

u \in L^{(r)} U_{S^{'}, T} \otimes Z [1 / 2]

such that

φ (λ_{Q} (u)) = Θ_{S, T}

Partial results in this direction had been obtained earlier by Burns [5] (including a

μ = 0

hypothesis and the assumption of the Gross-Kuz'min conjecture) and by Greither and Popescu [27] (including a

μ = 0

hypothesis and imprimitivity conditions on

S

Our results in [17] do not seem to directly imply the conjecture of Atsuta and Kataoka on Fitt

_{R} ({C l}^{T} (H)^{-})

or the conjecture of Burns. However, we prove an analogous result toward the latter, with

(S, T)

replaced by an alternate pair

(Σ, Σ^{'})

, in Theorem 5.6. This result turns out to be strong enough to deduce Theorem 4.3.

§ 6

we give a detailed summary of the proof of Theorem 5.6. Key ingredients are the

Z [G]

-modules

\nabla_{Σ}^{Σ^{'}} (H)

defined by Ritter and Weiss, and Ribet's method of using modular forms to construct Galois cohomology classes associated to

L

-functions.

4.3. Explicit formula for Brumer-Stark units

We conclude this section by describing a further direction in the study of BrumerStark units, that of explicit formulae and applications to explicit class field theory. This theme was initiated by Gross in [28] and [29] and developed by the first author and collaborators over a series of papers

[9, 10, 13, 15, 20]

Let

p

be as above and write

S^{'} = S \cup {p}

. Let

L

denote a finite abelian CM extension of

F

containing

H

that is ramified over

F

only at the places in

S^{'}

. Write

g = Gal (L / F)

and

Γ = Gal (L / H)

, so

g / Γ ≅ G

. Let

I

denote the relative augmentation ideal associated to

g

and

G

, i.e., the kernel of the canonical projection Aug:

Z [g] \to Z [G]

. Then

Θ_{S^{'}, T} (L / F)

lies in

I

, since its image under Aug is

\begin{matrix} (4.6) & Θ_{S^{'}, T} (H / F) = Θ_{S, T} (H / F) (1 - σ_{p}) = 0 \end{matrix}

Here

σ_{p}

denotes the Frobenius associated to

p

G

, and this is trivial since

p

splits completely in

H

. Intuitively, if we view

Θ_{S^{'}, T} (L / F)

as a function on the ideals of

Z [g]

, equation (4.6) states that this function "has a zero" at the ideal

I

; the value of the "derivative" of this function at

I

is simply the image of

Θ_{S^{'}, T} (L / F)

I / I^{2}

. Gross provided a conjectural algebraic interpretation of this derivative as follows. Denote by

{rec}_{P} : H_{B}^{*} \to Γ

the composition of the inclusion

H_{ß}^{*} ↪ A_{H}^{*}

with the global Artin reciprocity map

A_{H}^{*} \to Γ

Throughout this article we adopt Serre's convention [42] for the reciprocity map. Therefore

rec (ϖ^{- 1})

is a lifting to

G_{p}^{a b}

of the Frobenius element on the maximal unramified extension of

F_{p}

ϖ \in F_{p}^{*}

is a uniformizer.

Conjecture 4.4 (Gross, [29, CONJEcTURE 7.6]). Define

\begin{matrix} (4.7) & {rec}_{G} (u_{p}) = \sum_{σ \in G} ({rec}_{p} σ (u_{p}) - 1) {\tilde{σ}}^{- 1} \in I / I^{2} \end{matrix}

where

\tilde{σ} \in g

is any lift of

σ \in G

. Then

{rec}_{G} (u_{p}) \equiv Θ_{S^{'}, T}^{L / F} in I / I^{2}

The main result of [18] is the following.

Theorem 4.5. Let

p

be an odd prime and suppose that

p

lies above p. Gross's Conjecture 4.4 holds in

(I / I^{2}) \otimes Z_{p}

Our interest in this result is that by enlarging

S

and taking larger and larger field extensions

L / F

, one can use (4.7) to specify all of the

p

-adic digits of

u_{p}

. One therefore obtains an exact

p

-adic analytic formula for

u_{p}

. This formula can be described either using the Eisenstein cocycle or more explicitly via Shintani's method; the latter approach is followed in Section 7.2. In Section 7.3, we describe the argument using "horizontal Iwasawa theory" to show that Theorem 4.5 implies the conjectural exact formula. In Section 7.4 we summarize the key ingredients involved in the proof of Theorem 4.5, including an integral version of the Greenberg-Stevens

L

-invariant and an associated modified Ritter-Weiss module

\nabla_{L}

. In the setting of

F

real quadratic, Darmon, Pozzi, and Vonk have given an alternate, elegant proof of the explicit formula for the units

u_{p}

(Section 7.5). Their approach involves

p

-adic deformations of modular forms, rather than the tame deformations that we consider.

One significance of the exact formula is that we show that the collection of BrumerStark units, together with some easily described elements, generate the maximal abelian extension of the totally real field

F

Theorem 4.6. Let BS denote the set of Brumer-Stark units

u_{p}

as we range over all possible CMabelian extensions

H / F

and for each extension a choice of prime

p

that splits completely in

H

. Let

{α_{1}, \dots, α_{n - 1}}

denote any elements of

F^{*}

whose signs in

{\pm 1}^{n} / (- 1, \dots, - 1)

under the real embeddings of

F

form a basis for this

Z / 2 Z

-vector space. The maximal abelian extension of

F

is generated by

B S

together with

\sqrt{α_{1}}, \dots, \sqrt{α_{n - 1}}

F^{a b} = F (B S, \sqrt{α_{1}}, \dots, \sqrt{α_{n - 1}})

It is important to stress that the exact formula for

u_{p}

described in Sectin 7.2 can be computed without knowledge of the field

H

, using only the data of

F

, p, and the conductor of

H / F

. Furthermore, we can leave out any finite set of primes

p

without altering the conclusion of the theorem. In this way we obtain an explicit class field theory for

F

, i.e., an analytic construction of its maximal abelian extension

F^{ab}

using data intrinsic only to

F

itself. Explicit computations of class fields of real quadratic fields generated using our formula are provided in [17, Â§2.3] and [23].

In this section we recall various refinements of the strong Brumer-Stark conjecture.

5.1. The conjecture of Kurihara

In this section we describe the Fitting ideal of the minus part of the dual class group. For each

v

S_{ram}

, let

I_{v} \subset G_{v} \subset G

denote the inertia and decomposition groups, respectively, associated to

v

. Write

e_{v} = \frac{1}{# I_{v}} N I_{v} = \frac{1}{# I_{v}} \sum_{σ \in I_{v}} σ \in Q [G]

for the idempotent that represents projection onto the characters of

G

unramified at

v

. Let

σ_{v} \in G_{v}

denote any representative of the Frobenius coset of

v

. The element

1 - σ_{v} e_{v} \in Q [G]

is independent of choice of representative. Following [25], we define the Sinnott-Kurihara ideal, a priori a fractional ideal of

Z [G]

, by

{S K u}^{T} (H / F) = (Θ_{S_{\infty}, T}^{#}) \prod_{v \in S_{r a m}} (N I_{v}, 1 - σ_{v} e_{v})

Kurihara proved using the theorem of Deligne-Ribet and Cassou-NoguÃ¨s that

{S K u}^{T} (H / F)

is a subset of

Z [G]

(see [17, LEMMA 3.4]). The following conjecture of Kurihara is proven in

[17

, THEOREM 1.4].

Theorem 5.1. We have

{Fitt}_{R} ({C l}^{T} (H)^{-, v}) = {S K u}^{T} (H / F)^{-}

The definition of the Sinnott-Kurihara ideal is inspired by Sinnott's definition of generalized Stickelberger elements for abelian extensions of

Q

[44]. For a generalization of Sinnott's ideal to arbitrary totally real fields see [25, $2]. In general, Sinnott's ideal contains the Sinnott-Kurihara ideal but it may be strictly larger.

The plus part of the Sinnott-Kurihara ideal is not very interesting as the plus part of

Θ_{S_{\infty}, T}^{#}

is 0 . The plus part of the class group is much smaller than the minus part and seems harder to describe; for example, Greenberg's conjecture on the vanishing of lambda invariants implies that the order of the plus part is bounded up the cyclotomic tower. For abelian extensions of

Q

, the plus part is described by Sinnott using cyclotomic units.

5.2. The conjecture of Atsuta-Kataoka

It is, in fact, more natural to ask about the Fitting ideal of

{C l}^{T} (H)

, as opposed to the Pontryagin dual. A conjectural answer to this question has been provided in a recent paper of Atsuta-Kataoka [1] using the theory of shifted Fitting ideals developed by Kataoka [32]. We recall this notion now.

Let

M

be an

R

-module of finite length. Take a resolution

0 \to N \to P_{1} \to \dots \to P_{d} \to M \to 0

with each

P_{i}

of projective dimension

\leq 1

. Following [32], define the shifted Fitting ideal

{Fitt}_{R}^{[d]} (M) = (\prod_{i = 1}^{d} {Fitt}_{R} {(P_{i})}^{(- 1)^{i}}) {Fitt}_{R} (N)

The independence of this definition from the choice of resolution is proven in [32, THEOREM 2.6 AND PROPOSITION 2.7]. Let

g_{v} = 1 - σ_{v} + # I_{v} \in Z [G / I_{v}], h_{v} = 1 - e_{v} σ_{v} + N I_{v} \in Q [G]

Define the

Z [G]

-module

A_{v} = Z [G / I_{v}] / (g_{v})

Conjecture 5.2 (Atsuta-Kataoka). We have

{Fitt}_{R} ({C l}^{T} (H)^{-}) = (\prod_{w \in S_{r a m, H}} h_{v}^{-} {Fitt}_{R}^{[1]} (A_{v}^{-})) Θ_{S_{\infty}, T}

In [1], the authors give an explicit description of the ideal

h_{v}^{-} {Fitt}_{Z [G]^{-}}^{[1]} (A_{v}^{-})

appearing in Conjecture 5.2. Write

I_{v} = J_{1} \times \dots \times J_{s}

for cyclic groups

J_{i}, 1 \leq i \leq s

. For each

i

, put

N_{i} = N J_{i} = \sum_{σ \in J_{i}} σ \in Z [G]

Furthermore, put

d = \ker (Z [G] \to Z [G / G_{v}])

for the relative augmentation ideal. For each

1 \leq i \leq s

, put

Z_{i}

for the ideal of

Z [G]

generated by

N_{j_{1}} \dots N_{j_{s - i}}

, where

(j_{1}, \dots, j_{s - i})

runs through all tuples of integers satisfying

1 \leq j_{1} \leq \dots \leq j_{s - i} \leq s

. Define

I = \sum_{i = 1}^{s} Z_{i} χ^{i - 1}

Although

Z_{i}

depends on the decomposition of

I_{v}

into cyclic groups, the ideal

L

is independent (see [1, DEFINITION 1.2]).

Theorem 5.3 (Atsuta-Kataoka). We have

h_{v}^{-} {Fitt}_{Z [G]^{-}}^{[1]} (A_{v}^{-}) = (N I_{v}, (1 - e_{v} σ_{v}) L)

as fractional ideals of

Z [G]^{-}

Atsuta-Kataoka prove:

Theorem 5.4. The Equivariant Tamagawa Number Conjecture for

H / F

implies Conjecture 5.2.

5.3. The conjecture of Burns-Kurihara-Sano

The refinements mentioned above do not involve principal ideals. The method of Ribet, which attempts to show the inclusion of an arithmetically defined ideal into an analytically defined ideal, works well for principal ideals. From this point of view, it is natural to ask if there is an arithmetically defined object whose Fitting ideal is generated by the Stickelberger element

Θ_{S, T}

. Burns, Kurihara and Sano provided a conjectural answer to this question [6]. A modification of this statement (Theorem 5.6 below) is the main technical result in [17] from which all the results of Theorem 4.3 are deduced.

We now recall the statement of the conjecture of Burns-Kurihara-Sano. Let

H_{T}^{*}

be the group of

x \in H^{*}

such that

{ord}_{w} (x - 1) > 0

for each prime

w \in T_{H}

. Define

{Sel}_{S}^{T} (H) = {Hom}_{Z} (H_{T}^{*}, Z) / \prod_{w \notin S_{H} \cup T_{H}} Z

where the implicit map sends a tuple

(x_{w})

to the function

\sum_{w} x_{w} {ord}_{w}

. The

G

-action on

{Sel}_{S}^{T} (H)

is the contragradient

G

-action

(g φ) (x) = φ (g^{- 1} x)

Conjecture 5.5 (Burns-Kurihara-Sano). We have

{Fitt}_{R} ({Sel}_{S}^{T} (H)^{-}) = (Θ_{S, T}^{#})

We have proven a version of this result with altered sets

S

and

T

. Fix an odd prime

p

and put

R_{p} = Z_{p} [G]^{-}

. Define

Σ = S ∖ {v \in S : v ∤ p}

and

Σ^{'} = T \cup {v \in S : v ∤ p}

Theorem 5.6 ([17, THEOREM 3.3]). Let

{Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-} = {Sel}_{Σ}^{Σ^{'}} (H) \otimes_{Z [G]} R_{p}

. We have

{Fitt}_{R_{p}} ({Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-}) = (Θ_{Σ, Σ^{'}}^{#})

It turns out that Theorem 5.6 is strong enough to imply Kurihara's conjecture (Theorem 5.1). The key point is that there is a short exact sequence

\begin{matrix} (5.1) & 0 ⟶ {Sel}_{Σ}^{T} (H)^{-} ⟶ {Sel}_{Σ}^{Σ^{'}} (H)^{-} ⟶ \prod_{w \in S_{H}^{'}} {({(O_{H} / w)}^{*})}^{\lor, -} ⟶ 0, \end{matrix}

from which one deduces (see

[17

, THEOREM 3.7

]

)

{Fitt}_{R_{p}} ({Sel}_{Σ}^{T} (H)_{p}^{-}) = (Θ_{Σ, T}^{#}) \prod_{v \in S_{r a m}, v ∤ p} (N I_{v}, 1 - σ_{v} e_{v}) .

Since

{Sel}_{S_{\infty}}^{T} (H)^{-} ≅ {C l}^{T} (H)^{-, v}

, it then remains to calculate the effect of removing the primes

v \in S_{ram}, v ∣ p

from

Σ

. This is a delicate process using functorial properties of the Ritter-Weiss modules discussed in

§ 6

, and one obtains (see [17, APPENDIX B]) the desired result

{Fitt}_{R_{p}} ({Sel}_{S_{\infty}}^{T} (H)_{p}^{-}) = (Θ_{S_{\infty}, T}^{#}) \prod_{v \in S_{r a m}} (N I_{v}, 1 - σ_{v} e_{v})

6. RITTER-WEISS MODULES AND RIBET'S METHOD

In this section we summarize the proof of Theorem 5.6.

6.1. Ritter-Weiss modules

The

Z [G]

-module that shows up in our constructions with modular forms is a certain transpose of

{Sel}_{S}^{T} (H)

in the sense of Jannsen [31], denoted

\nabla_{S}^{T} (H)

. This module was originally defined by Ritter and Weiss in the foundational paper [40] without the smoothing set

T

. We incorporated the smoothing set

T

and established some additional properties of

\nabla_{S}^{T} (H)

in [17, APPENDIX A]. To describe these properties, we work over

R_{p} = Z_{p} [G]^{-}

and consider finite disjoint sets

Σ, Σ^{'}

satisfying the following:

$Σ \supset S_{\infty}$ and $Σ \cup Σ^{'} \supset S_{ram}$ .
$Σ^{'}$ satisfies the condition (2) on $T$ in Section 2.
The primes in $Σ^{'} \cap S_{ram}$ have residue characteristic $ℓ \neq p$ .

Note that the pair

(S, T)

from Section 2 and the pair

(Σ, Σ^{'})

considered in Section 5.3 both satisfy these conditions. The module

\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} = \nabla_{Σ}^{Σ^{'}} (H) \otimes_{Z [G]} R_{p}

satisfies the following:

There is a short exact sequence of $R_{p}$ -modules

\begin{matrix} (6.1) & 0 ⟶ {C l}_{Σ}^{Σ^{'}} (H)_{p}^{-} ⟶ \nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} ⟶ {(X_{H_{Σ}})}_{p}^{-} ⟶ 0 \end{matrix}

Here

{C l}_{Σ}^{Σ^{'}} (H)

denotes the quotient of

{C l}^{Σ^{'}} (H)

by the image of the primes in

Σ_{H}

Given a $R_{p}$ -module $B$ , a surjective $R_{p}$ -module homomorphism

\begin{matrix} (6.2) & \nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} \to B \end{matrix}

is equivalent to the data of a cocycle

κ \in Z^{1} (G_{F}, B)

and a collection of elements

x_{v} \in B

for

v \in Σ

satsifying the following conditions:

The cohomology class $[κ] \in H^{1} (G_{F}, B)$ is unramified outside $Σ^{'}$ , tamely ramified at $Σ^{'}$ , and locally trivial at $Σ$ .
The $x_{v}$ provide local trivializations at $Σ : κ (σ) = (σ - 1) x_{v}$ for $σ \in G_{v}$ .
The $x_{v}$ along with the image of $κ$ generate the module $B$ over $R_{p}$ .

The tuples

(κ, {x_{v}})

are taken modulo the natural notion of coboundary, i.e.,

(κ, {x_{v}}) \sim (κ + d x, {x_{v} + x})

for

x \in B

The module $\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}$ has a quadratic presentation, i.e., there exists an exact sequence of $R_{p}$ -modules

\begin{matrix} (6.3) & M_{1} \overset{A}{\to} M_{2} ⟶ \nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} ⟶ 0 \end{matrix}

where

M_{1}

and

M_{2}

are free of the same finite rank.

The module $\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}$ is a transpose of ${Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-}$ , i.e., for a suitable quadratic presentation (6.3) of $\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}$ , the cokernel of the induced map

\begin{matrix} (6.4) & {Hom}_{R_{p}} (M_{2}, R_{p}) \overset{A^{T, #}}{\to} {Hom}_{R_{p}} (M_{1}, R_{p}) \end{matrix}

is isomorphic to

{Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-}

. Here we follow our convention of giving Hom spaces the contragradient

G

-action.

The quadratic presentation property (6.3) implies that

{Fitt}_{R_{p}} (\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}) = \det (A)

is principal. The transpose property (6.4) implies that

\begin{matrix} (6.5) & {Fitt}_{R_{p}} (\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}) = {Fitt}_{R_{p}} {({Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-})}^{#} \end{matrix}

Theorem 5.6 is therefore equivalent to

\begin{matrix} (6.6) & {Fitt}_{R_{p}} (\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}) = (Θ_{Σ, Σ^{'}}) \end{matrix}

We now fix

(Σ, Σ^{'})

to be the pair defined in Section 5.3. In the remainder of this section we outline how (6.6) is proved using Ribet's method. Throughout, an unadorned

Θ

denotes

Θ_{Σ, Σ^{'}} (

and

Θ^{#}

denotes

Θ_{Σ, Σ^{'}}^{#}

6.2. Inclusion implies equality

An interesting feature of Ribet's method is that it tends to prove an inclusion in one direction, that of an algebraically defined ideal contained within an analytically defined ideal. In our setting, we use it to prove

\begin{matrix} (6.7) & {Fitt}_{R_{p}} (\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}) \subset (Θ), equivalently, {Fitt}_{R_{p}} ({Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-}) \subset (Θ^{#}) \end{matrix}

We then employ an analytic argument to show that this inclusion is an equality. It is important to note that the inclusion (6.7) is the reverse direction of that required by the Brumer-Stark and Strong Brumer-Stark conjectures. It is therefore essential for our approach that one actually has the statement of an equality rather than just an inclusion (and an analytic argument to deduce the equality from the reverse inclusion). For this reason, the conjecture of Burns stated in Section 5.3 (more precisely the analog of it stated in Theorem 5.6) plays an essential role in our strategy.

Let us describe the analytic argument in the special case

Σ = S_{\infty}

, i.e. there are no primes above

p

ramified in

H / F

. In this case

\begin{matrix} (6.8) & {Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-} ≅ {C l}^{Σ^{'}} (H)_{p}^{-} \end{matrix}

is finite and

Θ

is a non-zero-divisor. Using (6.7), write

\begin{matrix} (6.9) & {Fitt}_{R_{p}} ({Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-}) = (Θ^{#} \cdot z) for some z \in R_{p} \end{matrix}

We must show that

z \in R_{p}^{*}

. An elementary argument (see [17, $2.3]) shows that (6.9) implies

\begin{matrix} (6.10) & # {Sel}_{Σ}^{Σ^{'}} (H)_{p}^{-} = {(\prod_{\begin{matrix} χ \in \hat{G} \\ χ odd \end{matrix}} χ (Θ^{#} \cdot z))}_{p} \end{matrix}

where the subscript

p

on the right denotes the

p

-power part of an integer. Yet the analytic class number formula implies (see

[17, $ 2.1]

)

\begin{matrix} (6.11) & \prod_{\begin{matrix} χ \in \hat{G} \\ χ odd \end{matrix}} χ (Θ^{#}) = \prod_{\begin{matrix} χ \in \hat{G} \\ χ odd \end{matrix}} L_{Σ, Σ^{'}} (χ, 0) ≐ # {C l}^{Σ^{'}} (H)^{-} \end{matrix}

where

≐

denotes equality up to a power of 2 . Combining (6.8), (6.10), and (6.11), one finds that

χ (z)

is a

p

-adic unit for each odd character

χ

. It follows that

z \in R_{p}^{*}

as desired.

The generalization of this argument to arbitrary

Σ

requires a delicate induction and is described in

[17, $ 5]

6.3. Ribet's method

We now describe our implementation of Ribet's method to prove the inclusion (6.7). The idea is to use the Galois representations associated to Hilbert modular forms to construct an

R_{p}

-module

M

and a surjection

\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-} \to M

such that

{Fitt}_{R_{p}} (M) \subset (Θ)

. The properties of Fitting ideals imply that (6.7) follows from the existence of such a surjection. As described in (6.2), a surjection from

\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}

can be constructed by defining a cohomology class

[κ] \in H^{1} (G_{F}, M)

satisfying certain local conditions along with local trivializations at the places in

Σ

Ribet's method was described in great detail by Mazur in a beautiful article written for the celebration of Ribet's 60th birthday [36]. We borrow from this the following schematic diagram demonstrating the general path one follows to link

L

-values (in our case, the Stickelberger element

Θ

) to class groups (in our case, the Ritter-Weiss module

\nabla_{Σ}^{Σ^{'}} (H)_{p}^{-}

Let us now trace this path in our application.

6.3.1. $L$ -functions to Eisenstein series

The key connection between

L

-functions and modular forms in Ribet's method is that

L

-functions appear as constant terms of Eisenstein series. We now define the space of modular forms in which our Stickelberger element

Θ

appears.

Let

k > 1

be an integer such that

k \equiv 1 (mod (p - 1) p^{N})

for a large value of

N

. Let

n \subset O_{F}

denote the conductor of

H / F

. Let

M_{k} (n; Z)

denote the group of Hilbert modular forms for

F

of level

n

with Fourier coefficients in

Z

. For each odd character

χ

G

valued in

C_{p}^{*}

, let

M_{k} (n, χ) \subset M_{k} (n; Z) \otimes C_{p}

denote the subspace of forms of nebentypus

χ

. Let

χ : G_{F} \to G \to R_{p}^{*}

denote the canonical character, where the first arrow is projection and the second is induced by

G ↪ Z [G]^{*}

Definition 6.1. The space

M_{k} (n, χ; R_{p})

of group-ring valued Hilbert modular forms of weight

k

and level

n

over

R_{p}

consists of those

f \in M_{k} (n; Z) \otimes R_{p}

such that

χ (f) \in M_{k} (n, χ)

for each odd character

χ

. Let

S_{k} (n, χ; R_{p})

denote the subspace of cusp forms. We define

M_{k} (n, χ; Frac (R_{p}))

and

S_{k} (n, χ; Frac (R_{p}))

similarly.

Hilbert modular forms

f

are determined by their Fourier coefficients

c (m, f)

indexed by the nonzero ideals

m \subset O_{F}

and their constant terms

c_{λ} (0, f)

indexed by

λ \in {C l}^{+} (F)

, the narrow class group of

F

. For odd

k \geq 1

, there is an Eisenstein series

E_{k} (χ, 1) \in M_{k} (n, χ)

whose Fourier coefficients are given by

c (m, E_{k} (χ, 1)) = \sum_{\begin{matrix} a ∣ m \\ (m / a, n) = 1 \end{matrix}} χ (m / a) {N a}^{k - 1}

2.2. The approach of Jacquet-Rallis

3. COMPARISON: LOCAL TRANSFER AND FUNDAMENTAL LEMMA

4. GLOBAL ANALYSIS OF JACQUET-RALLIS TRACE FORMULAE

5. LOOKING FORWARD

ACKNOWLEDGMENTS

FUNDING

REFERENCES

RAPHAÃ‹L BEUZART-PLESSIS

THE COHOMOLOGY OF SHIMURA VARIETIES WITH TORSION COEFFICIENTS

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

2. A VANISHING CONJECTURE FOR LOCALLY SYMMETRIC SPACES

3. THE HODGE-TATE PERIOD MORPHISM

4. COHOMOLOGY WITH MOD ℓ â„“ â„“\ellâ„“ COEFFICIENTS

5. COHOMOLOGY WITH MOD p p ppp AND p p ppp-ADIC COEFFICIENTS

6. APPLICATIONS BEYOND SHIMURA VARIETIES

ACKNOWLEDGMENTS

FUNDING

REFERENCES

ANA CARAIANI

ON THE BRUMER-STARK CONJECTURE AND REFINEMENTS

SAMIT DASGUPTA AND MAHESH KAKDE

Abstract

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. BACKGROUND AND MOTIVATION

2. STARK'S CONJECTURE

3. STARK'S CONJECTURES AT FINITE PLACES

4. THE BRUMER-STARK CONJECTURE

4.1. Annihilation of class groups

4.2. Our results

4.3. Explicit formula for Brumer-Stark units

5. REFINEMENTS OF STARK'S CONJECTURE

5.1. The conjecture of Kurihara

5.2. The conjecture of Atsuta-Kataoka

5.3. The conjecture of Burns-Kurihara-Sano

6. RITTER-WEISS MODULES AND RIBET'S METHOD

6.1. Ritter-Weiss modules

6.2. Inclusion implies equality

6.3. Ribet's method

6.3.1. L L LLL-functions to Eisenstein series

4. COHOMOLOGY WITH MOD $ℓ$ COEFFICIENTS

5. COHOMOLOGY WITH MOD $p$ AND $p$ -ADIC COEFFICIENTS

6.3.1. $L$ -functions to Eisenstein series